Tilburg University Texas Hold’em van der Genugten, B.B ... · Texas Hold’em: a game of skill Ben van der Genugten and Peter Borm CentER and Department of Econometrics and OR,

Tilburg University

Texas Hold’em

van der Genugten, B.B.; Borm, P.E.M.

Document version:Early version, also known as pre-print

Publication date:2014

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Citation for published version (APA):van der Genugten, B. B., & Borm, P. E. M. (2014). Texas Hold’em: A Game of Skill. Tilburg: Department ofEconometrics.

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1

Texas Hold’em: a game of skill

Ben van der Genugten and Peter Borm

CentER and Department of Econometrics and OR, Tilburg University, Tilburg, The Netherlands

Abstract. On the basis of the detailed mathematical analysis of realistic approximations of Texas

Hold’em cash games and tournaments as provided in [1] and [2], respectively, this paper reconfirms

the findings of the court in The Hague [3] that on the basis of the methodology of relative skill it can

be concluded that Texas Hold’em in all its practical variants is a game of skill.

1. Introduction: on the methodology of relative skill.

The distinction between games of chance and games of skill is a prominent topic due to the rules set

out in the gaming acts of various countries. This paper will focus on this distinction within the

framework of the Dutch gaming act. Since different gaming acts typically have the same

characteristics our conclusions however do not only apply to the situation in the Netherlands but can

be applied in a rather broad and general sense. The game under consideration here is a specific

variant of poker: Texas Hold’em. Especially this variant of poker has gained worldwide popularity

over the past ten years, not only in casinos and on internet but also as the poker game played and

advertised in championships broadcasted on television throughout the world.

Roughly speaking the Dutch gaming act states that is not allowed to exploit, without license, games

with monetary payoffs in which the participants do not have a predominant influence on their game

results. These games are called games of chance. Since influencing game results requires skill

(studying, experience, counting cards, training memory, observational skills, etc.) other games are

called games of skill. Consequently, each game with monetary payoffs involving chance elements

(drawing cards, spinning of a wheel, throwing dice, etc.) in principle can be classified either as a game

of chance or as a game of skill. Of course the crucial word in the gaming act is predominant, which

means that the influence of skill should be weighed against the influence of chance.

The influence of skill is referred to as the learning effect LE in a game. Thus, the learning effect

measures the maximal difference in game result directly due to the influence or skill of players

themselves. It can be represented by the additional gains a player is able to obtain in evolving from a

serious but inexperienced beginner into a well experienced and established player who has mastered

all intricacies of the game: LE ≥ 0. In fact, empirical observations of real game play could provide valid

indications of the absolute size of the learning effect.

Regarding cash poker and tournament poker, a wide variety of books (cf. [4], [5], [6] and [7]), articles

(cf. [8] and [9]), and internet sites (cf. [10]) exclusively focus on the learning effect and, in particular,

on improving actual play by creating a player’s awareness by discussing general game aspects and

providing tips and tricks, by devising and analyzing theoretically strong poker strategies and/or by

analyzing expert play in practice. None of these studies however touch the issue of the influence of

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an expert strategy to be predominant in a methodological way. For this reason these studies have

and will be ignored in the legal discussion of how to qualify Texas Hold’em according to a gaming act.

To adequately measure if players are able to execute predominant influence on their game result,

the learning effect should rather be considered in a relative sense. For this reason we compare the

learning effect with the so-called random effect RE that measures to what extent a skilful and

advanced player is restricted in further improving his gains by the presence of the chance elements.

It can be represented by the additional gains an experienced player would be able to obtain in the

fictive situation that he would know (but cannot influence) the outcomes of all possible chance

elements in advance before having to decide on his actions in the game: RE ≥ 0. It will be clear that

empirical observations of game play will not provide information about the absolute size of the

random effect. The random effect has to be calculated using an explicit and tailor-made analysis on

the basis of the rules and specifications of the game at hand. In a controlled experimental setting

however information on the random effect can be obtained as is shown and documented for the

specific case of Texas Hold’em in [11].

Having obtained the two numbers LE and RE in monetary units, the relative level of skill S is defined

as the quotient of LE and LE+RE: S = LE / (LE+RE). We want to note that games in which both the

learning effect and the random effect equal 0 are of no practical interest and are therefore left out of

consideration. It will be clear that the level of relative skill S is a number between 0 and 1.

A game with LE=0, i.e., if no player can improve upon the results of a beginner, has a level of relative

skill equal to 0. Such games are called pure games of chance. On the other hand, a game with RE=0,

i.e., if the chance elements put no restriction the results of a skilful player, has a level of relative skill

equal to 1. Such games are called pure games of skill. Although there are many different types of

games, all with different characteristics regarding game material, chance generators, number of

players, complexity etc., the methodology of relative skill can in principle be applied to any strategic

game where a player’s influence on his game results is through the quality of his strategy selection

only. A more detailed discussion on the definitions of learning effect, random effect, level of relative

skill and strategic games can be found in [12] and [13]. The way how to exactly quantify the level of

relative skill is elaborated upon in the next section.

The above described methodology of relative skill has shown its value as an operational tool in

practice and, moreover, turned out to be decisive in courts of law. The application of the

methodology of relative skill is flexible and mature in the sense that it requires a tailor-made

quantitative analysis for each game separately although the basic qualitative ideas and the

underlying conceptual framework have remained the same in all specific cases at hand throughout

the years. It has been not only been applied to one-person games as Roulette (S=0 for French

roulette), Caribbean stud poker (S= 0.003), and Blackjack (S =0.05 in Holland Casinos), and many,

many more, but also to more-person games with strong interaction between the players like cash

and tournament poker variants. For a partial overview on levels of relative skill for various types of

games we refer to [14]. Importantly the methodology of relative skill has played a decisive role in

court. The most recent example of this type can be found in [3] in which the court of The Hague

integrally accepts the methodology of relative skill and its underlying conceptual reasoning as

presented in [2] and concludes that the Texas Hold’em cash games and tournaments under

consideration could not be viewed as games of chance. At an earlier stage the court of Arnhem [15]

3

ruled on the basis of [16] and [17] that particular types of interactive more-person management

games like Grand Prix manager and Competition manager, with levels of relative skill approximately

equal to 0.3, should not be considered as games of chance.

As its main contribution the methodology of relative skill offers an objective and mathematical tool

to order and compare different types of strategic games on the basis of well-defined relevant

concepts and terminology and consistent reasoning. Moreover, it is legally constructive in the sense

that it offers the possibility to set a legal skill threshold level for the level of relative skill above which

a game should be considered a game of skill. Allowing for consistent legislation on the basis of the

ruling [15] on management games and the fact that no variant of the (legally classified) game of

chance Blackjack has a level of relative skill above 0.2, as is seen in [18] and [19], the skill threshold

level should be set between 0.2 and 0.3. In particular, if for a specific game the level of relative skill

well exceeds 0.3, then this game should be legally classified as a game of skill.

Due to the conceptual choices underlying the computation of the learning effect and the random

effect the methodology of relative skill has one additional advantage, especially in games with

several interacting players like Texas Hold’em. It requires the input of only one subjective parameter:

the strategic behaviour of a beginner. In this choice regarding the behaviour of a beginner the social

impact of the game and the general level of the participants in practice can be incorporated.

Moreover, the choice can also be based on statistical empirical evidence or professional knowledge.

This helps to better focus the discussion about the skill level of a game to its core features. From a

methodological perspective this input parameter allows for a sensitivity analysis of relative skill when

varying a particular choice of the strategy of a beginner possibly based on the development of the

game as played in practice over time.

In our mathematical analysis of realistic variants of Texas Hold’em the behaviour of a beginner is

selected carefully. Generally speaking, the format of our beginner’s strategy is generally applicable to

a wide range of complex Poker variants and therefore can already be considered as advanced.

Moreover, given the advanced format, our choice of the concrete parameters defining a beginner’s

strategy for real-life variants of Texas Hold’em is derived from extrapolating the choice made from

several options for a basic Texas Hold’em variant on the basis of having the lowest level of relative

skill. In this sense the approximated levels of relative skill of Texas Hold’em variants can be viewed as

underestimates for the real levels of relative skill.

2. Implementing relative skill

To formally implement the methodology of relative skill for a specific game via the general concepts

of learning effect and random effect described above, we consider three types of players: the

beginner, the optimal player and the fictive player.

The beginner with expected monetary result R(0) models a serious but inexperienced player who has

just mastered the rules of the game and who is endowed with a fixed particular (and typically naive)

strategy which is the methodology’s main choice parameter. Here (and elsewhere) the word

expected is used in the strict sense of probability theory, calculated on the basis of the chance

elements present in the game at hand. These chance elements can be induced both externally (e.g.

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by the dealing of cards) and internally (by randomization imposed by the players themselves in taking

decisions).

The optimal player with expected monetary result R(m) models an expert player who completely

masters the rules and intricacies (in Texas Hold’em in particular the strategic aspects of bluffing and

sandbagging) of the game to maximize his monetary gains.

The fictive player with expected monetary result R(f) models an expert player who on top of his

knowledge and expertise will be informed about the realizations of all chance elements (of both

external and internal type) and who can adequately use this additional information to further

increase his monetary gains.

After having chosen a fixed beginner’s strategy the monetary results R(0), R(m) and R(f) are

unambiguously defined for one-player games with no strategic interaction between participants like

Roulette and Blackjack. For more-person games with strategic interaction between the participants

like Texas Hold’em there are several complications to deal with. Not only can the game involve

different player roles but typically the monetary result of each player type in a certain role will also

depend on the strategic behaviour of its opponents in the game. To allow for a consistent analysis,

for each player role we will calculate the monetary results of the three player types assuming that all

opponents are beginners and therefore endowed with the fixed strategy of a beginner (which may

depend on the specific role of such a player), while thereafter we will take the appropriate mean

over the player roles to get the monetary game results R(0), R(m) and R(f) of the three player types.

Subsequently we define the learning effect by LE= R(m)- R(0) and the random effect by RE=R(f)- R(m).

As a consequence the level of relative skill S = LE / (LE + RE) can be rewritten as

S = (R(m) – R(0)) / (R(f) – R(0))

3. On the level of relative skill of Texas Hold’em cash games

The more qualitative analysis of several poker variants provided in [20] concludes that Texas Hold’em

in all its cash variants should not be classified as a game of chance on the basis of the methodology of

relative skill. At this point in time we are able to offer further quantitative evidence for this

conclusion based on explicit computations of the level of relative skill for more realistic variants of

Texas Hold’em. For expositional purposes we will only consider two main variants here: fixed limit

and no limit Texas Hold’em. The document “Texas Hold’em cash games; a mathematical analysis”[1]

contains detailed descriptions of all variants under consideration, the computational set-up and an

extensive overview of the computer output regarding relative skill, also for pot limit and spread limit

variants, including possible variations in measuring a player type’s game result.

As argued before, the central issue in applying the methodology of relative skill is the selection of an

appropriate strategy of a beginner in each possible player role. In our quantification of a strategy of

the beginner’s type we are primarily led by general observations in the popular and more scientific

literature on poker and the findings in the Texas Hold’em experiment reported on in [3]. Roughly

speaking it can be concluded that a beginner in Texas Hold’em does not discriminate between roles

(positions at the table) and will base his decision at any time moment solely on his own (preflop)

5

cards, the available community cards (if applicable), the current pot size and the number of

opponents left. These considerations lead to a general parametrized format of a beginner’s strategy

applicable to all poker variants. To determine adequate values for the various parameters we used

an extrapolation procedure from the choices made for a basic cash poker variant. To motivate these

choices, we considered four possible options for a beginner’s strategy in the basic variant and, to be

on the safe side, we selected the option with the lowest relative level of skill.

A standard fixed limit game of Texas Hold’em typically is played with a complete deck of cards, hosts

between two to ten players, consists of 4 rounds (pre-flop, flop, turn, river), has a bet equal to two

times the ante, allows for a maximum of 3 raises (excluding bet) per round, and has no possibility of

all-in. From these aspects we selected 3 basic variables: the number of players, the number of rounds

and the maximum number of raises per round. Setting the number of players equal to 2, the number

of rounds equal to 2, and the maximum number of raises per round equal to 1 (while adopting the

remaining standard settings), our basic variant of Texas Hold’em is obtained as the starting point of

our analysis. Although less complex (but more tractable), this basic variant contains all important

strategic ingredients of Texas Hold’em itself.

At this point we briefly want to comment on the accuracy of the levels of relative skill derived below.

Our statistical approach will not use a random sample since, due to the characteristics of an optimal

an a fictive player type, the game result of an optimal player will be positively biased towards the

game result of a fictive player, requiring a random sample which from a computational perspective is

too large. This problem is circumvented in our analysis by using a different technique of

representative sampling as described in some detail in [1] to reduce the bias. The resulting accuracy

of the derived levels of relative skill is approximately 0.02.

We now present a compact overview of the quantitative findings obtained in [1]. To maintain a clear

focus, this overview is incomplete. The level of relative skill of the basic variant is at least 0.29. The

marginal impact of varying one of our three parameters while keeping the other two at the basic

level does not lower the level of relative skill. In particular, allowing for 2 or 3 raises per round, the

level of relative skill remains about the same. If, on the other hand, one increases the number of

players to 3, the level of relative skill increases substantially to 0.42, while, if one increases the

number of rounds to 4 the level of relative skill increases substantially too, in this case to 0.37.

Furthermore, a similar type of analysis in a Texas Hold’em no limit setting leads to a level of relative

skill of at least 0.52 for the basic variant while identical marginal effects as above can be observed

when varying the number of possible raises per round, the number of players and the number of the

rounds.

The overall conclusion is that the relative level of skill of Texas Hold’em cash variants will well exceed

the skill threshold level which, as argued before, should be set between 0.2 and 0.3.

4. On the level of relative skill for Texas Hold’em tournaments

Texas Hold’em tournaments are rapidly gaining popularity. Typically in such a tournament monetary

prizes are awarded on the basis of a final ranking on all participants as determined by the results (in

fiches or points) during several tournament rounds (knock-out or not) of the cash version of Texas

6

Hold’em. As argued in [2], almost any casino cash game can be organized in the form of tournament

with this casino game played in its various tournament rounds and moreover, that it is not clear a

priori if a tournament should be automatically legally classified in the same way as its underlying cash

variant. In particular, for the case of Texas Hold’em tournaments, the statement “since in each round

one plays a game of skill, the tournament itself is a game of skill too”, cannot be justified at all

without an explicit analysis of relative skill. The final qualification of a tournament will heavily

depend on the number of participants in relation to the number of tournament rounds (relatively

short knock-out systems and lengthy full competitions will differ in relative skill) and especially on the

exact prize scheme (relative skill will substantially differ in a prize scheme with one winner from a

prize scheme with a relatively flat prize scheme for e.g. the top 10%). In this sense the analysis of

tournaments has definite similarities with the study on management games as presented in [16] and

[17].

In [2] a detailed, tailor-made analysis has been made for concrete set-ups for Texas Hold’em

tournaments. From this analysis the general conclusion can be drawn that the level of relative skill of

Texas Hold’em tournaments as organized in the Netherlands varies between 0.3 and 0.5, depending

on the exact tournament specifications.

5. Conclusion

The mathematical analysis of realistic variants of both cash versions and tournament versions of

Texas Hold’em indicate levels of relative skill that well exceed the skill threshold level. It confirms

earlier more qualitative findings that Texas Hold’em in all its practical variants should be considered

as games of skill, based on a consistent application of the methodology of relative skill.

References

[1] van der Genugten, B. and P. Borm (2012). The skill of Texas Hold’em cash games: a mathematical

analysis. Available at

http://www.tilburguniversity.edu/webwijs/show/?uid=p.e.m.borm&uid=p.e.m.borm

[2] Borm. P. and B. van der Genugten (2009). Cash and tournament poker: games of skill. Report,

Department of Econometrics and OR, Tilburg University, Tilburg, The Netherlands (In Dutch).

Available at http://www.tilburguniversity.edu/webwijs/show/?uid=p.e.m.borm&uid=p.e.m.borm

[3] http://www.rechtspraak.nl/Pages/default.aspx (2011). Ruling about organizing poker

tournaments, national jurisprudence number LJN: BN 0013, Rechtbank ‘s Gravenhage, 09/867520-08.

[4] Suzuki, S. and P. Cizmar (1998). Poker tournament strategies. Two plus Two Publishing,

Henderson.

[5] Sklansky, D. (2004). The theory of poker. Two plus Two Publishing, Henderson, 6th printing.

[6] Sklansky, D. (2002). Tournament poker for the advanced player. Two plus Two Publishing,

Henderson.



http://www.rechtspraak.nl/Pages/default.aspx

7

[7] Graig, M. (2007). The full tilt poker strategy guide: tournament edition. Grand Central Publishing,

New York.

[8] Levitt, S.D. and T.J. Miles (2011). The role of skill versus luck in poker: evidence from the world

series of poker. Report, Department of Economics, University of Chicago, USA.

[9] Levitt, S.D., Miles, T.J., and A.W. Rosenfield (2012). Is Texas Hold’em a game of chance? A legal

and economic analysis. Report, Department of Economics, University of Chicago, USA.

[10] Some websites on how to play poker tournaments: http://www.mosesbet.com/mtt-strategy,

http://www.playwinningpoker.com/poker/tournaments, http://www.tightpoker.com/tournaments,

http://www.pokertournamentstrategy.org/

[11] Maaten, R., Borm, P., van der Genugten, B., and R. Hendrickx (2010). The relative skill of Texas

Hold ‘Em: an experiment in cooperation with VARA Nieuwslicht. Report, Department of Econometrics

and OR, Tilburg University, Tilburg, The Netherlands (In Dutch). Available at


[12] Dreef, M., Borm, P., and B. van der Genugten (2004). A new relative skill measure for games with

chance elements. Managerial and Decision Economics, 25, 255-264.

[13] Hendrickx, R., Borm. P., and B. van der Genugten (2008). Measuring skill in more-person games

with applications to poker. CentER Discussion paper 2008-106, Tilburg University, Tilburg, The

Netherlands.

[14] van der Genugten, B., Das. M., and P. Borm (2001). Gambling skillfully in casinos. Academic

Service, Schoonhoven, The Netherlands, ISBN 90 395 1571 9 (In Dutch).

[15] http://www.rechtspraak.nl/Pages/default.aspx (2005). Ruling about organizing management

games, national jurisprudence number LJN: AS6590, Rechtbank Arnhem, 105364.

[16] van der Genugten, B., Borm, P., and M. Dreef (2004). The application of the Dutch gaming act on

the management games Competition manager and Grand Prix manager. Report, Department of

Econometrics and OR, Tilburg University, Tilburg, The Netherlands (In Dutch).

[17] van der Genugten, B., Borm, P., and M. Dreef (2005). The application of the Dutch Gaming Act

on the management games Competition Manager and Grand Prix manager (continued). Report,

CentER of Applied Research, Tilburg University, Tilburg, The Netherlands (In Dutch).

[18] van der Genugten, B. (1993). Blackjack in Holland Casinos: how to beat the dealer. Tilburg

University Press, Tilburg, The Netherlands. ISBN 90 361 9793 7.

[19] van der Genugten, B., Das, M., and P. Borm (1999). An analysis of Blackjack variants. Report,

CentER of Applied Research, Tilburg University, Tilburg, The Netherlands, by the direction of Moulin

Rouge Casino- und Gastronomiebetriebsgesellschaft M.B.H., Vienna (In Dutch).

[20] Borm, P. and B. van der Genugten (2005). Poker: a game of skill! Report, Department of

Econometrics and OR, Tilburg University, Tilburg, The Netherlands, by the direction of Concord Card

Casino, Vienna (In German).

http://www.mosesbet.com/mtt-strategy

http://www.playwinningpoker.com/poker/tournaments

http://www.tightpoker.com/tournaments

http://www.pokertournamentstrategy.org/


http://www.rechtspraak.nl/Pages/default.aspx

8

THE SKILL OF TEXAS HOLD’EM CASH GAMES: A MATHEMATICAL

ANALYSIS

1. INTRODUCTION

In this paper we give the specific details about THM-cashgames, showing that their skill is larger than

the juridical threshold, somewhere between 0.2 and 0.3. The general methodology about relative

skill and the result of its application to THM is described in [1]. This paper gives a precise description

of the calculations as far as it concerns cash games. In the following we suppose that the rules of

THM are known. Therefore, for shortness, we refer hereafter to specific rules using only some

catchwords.

A THM-cashgame consists of 4 rounds: preflop, flop, turn, river (MaxRounds = 4). Betting starts with

a fixed Small Blind (SB) and Big Blind (BB). The real decisions of players can be Fold, Call, Check, Bet

and Raise, dependent on the stage of the game. The maximum number of raises per round is 3

excluding the bet (MaxRaise = 3). The bet amount is doubled from the turn on (DoubleRound = 3). If

at a certain stage of the game the chips of player are insufficient for a certain decision he can go all-

in. The number of players varies: 2 <= Players <= 10. Betting rules can vary: fixed-, spread-, pot- and

no-limit betting. We refer to games with all these rules as full games.

Full games have very large game trees and are too difficult to analyze with a direct approach.

Therefore in section 1 we start with a simplified fixed-limit game with Players = 2, MaxRounds = 2

(preflop, river), DoubleRound = 2 and with MaxRaise = 1. We refer to this game as the basic game.

This basic game contains all the specific strategic possibilities of full games. It is extensively analyzed

in detail by varying beginner strategies, looking especially for cases with a low skill for comparison

with the juridical threshold.

Thereafter, in section 2 we consider fixed-limit generalizations by varying separately Players,

MaxRounds and MaxRaise towards usual values, again paying attention to cases with low skill. From

these results we extrapolate conclusions to cases where these parameters are varied simultaneously.

9

We extend the fixed-limit basic games of section 1 in section 3 to spread-limit, in section 4 to pot-

limit and in section 5 to no-limit. Again the emphasis is on variants with low skill. This leads to the

final conclusion that the skill of all full versions of cash-THM are larger than the juridical threshold.

THM is played with a simple card-deck of 52 cards. But this leads to a tremendous number of card

distributions over players and community cards. So the analysis cannot be based on this population

but, necessarily, on samples. This raises special questions about dispersion and bias (see section 1.2).

For the analysis of cash-THM a computer program is developed under Matlab. It covers all variations

mentioned above by just filling in the appropriate parameters (see Appendix A2). But this variety has

only theoretical value. In practice the resulting game trees and sample sizes are always restricted by

computer time and computer memory. For the obtained results in this paper the calculations were

done with the 64-bit version of Matlab under a Windows-7 PC ( 64-bit, i7-processor) with an internal

memory of 16 Gb.

1. THE BASIC GAME (SIMPLIFIED FIXED-LIMIT THM)

1.1 Rules and game tree

The global rules are described in the introduction. The game tree of figure 1.1.1. reveals all details:

- no all-in situations: the start capital (amount of chips) of the players is always large

enough),

- maximal 1 raise per round (exclusive the bet) means no reraise,

- the BB-player in the first round gets always has a real move (see node 3 in the figure);

his decision Bet is sometimes called Raise and is always counted as a raise,

- after the first round (preflop) the last round comes immediately (the river with 5 cards),

- the bet is doubled in the last round (DoubleRound = 2),

- 2 players (headsup): player I is the SB, player II the BB,

- as usual with headsup, player I starts the betting in the first round, player II in the last

round.

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In the figure the decisions and the corresponding betting chips are indicated at the leaves, the

amounts of betting chips of the 2 players are indicated at the nodes. The nodes themselves are

numbered in such a way that from the top on a higher node is never followed by a lower node.

Furthermore, the numbers of a specific round are always greater than the numbers of a foregoing

round. The numbers of endnodes are put within squares, the intermediate ones within circles. At

each intermediate node the corresponding player and the pot composition is indicated; the leaves

show the decisions and the corresponding bets.

11

12

Figure 1.1.1 The game tree of the basic game.

We have included this figure because it makes immediately clear the specific rules of the basic game.

Appendix A3 contains all details as they are generated by the computer program.

1.2 Sampling from the population of card distributions

Even with 2 players and 2 rounds (the basic game) the population of card distributions is too large to

work with:

C(52,2) x C(50,2) x C(48,5) = 2 781 381 002 400 .

We have to take a representative sample and act as if this is the whole population. It should be

sufficiently large for the desired accuracy of the skill ( 0.02, say).

A special problem for calculations with respect to the optimal player is that the gain result has a

positive bias (towards the fictive player) for any sample. However, the bias for a random sample can

be reduced considerably by taking a representative sample in a special way.

In section 1.3 we give the results for various beginner strategies. In every subsection we use at first a

very elementary sample of size 4 and give the detailed calculations which can be controlled by hand.

Thereafter, we use a representative sample of size 3 288 600, sufficiently large for getting the desired

accuracy. Of course results follow now from the computer program.

For sake of comparison with different rules these elementary and representative samples are

repeatedly used whenever the number of rounds and the number of players is 2. The samples

themselves are described in the sections 1.2.1 and 1.2.2.

1.2.1 Elementary sample

The elementary sample is shown in table 1.2.1.1. Cards are numered in the following way:

1 = 2c(lubs), 2 = 2d(iamonds), 3 = 2h(earts) , 4 = 2s(pades), . . . , 52 = A(spades).

CardComb Preflop I Preflop II River

1

2

1 8

1 8

42 46

42 46

26 36 43 48 49

3 4 5 33 38

13

3

4

42 46

42 46

1 8

1 8

26 36 43 48 49

3 4 5 33 38

Table 1.2.1.1 Card numbers of the elementary sample of 4 card combinations.

So in our analysis we act as if this set forms the whole population of card distributions. We do this

just for illustrating the concepts with detailed calculations. It is even not a random sample but

contrarily carefully chosen. We see that for each player there are only 2 preflops and also only 2

rivers. This leads to 4 carefully mixed card combinations. This specific small sample makes results not

completely trivial. Note that the knowledge of the optimal player is almost the same as that of the

fictive player. The difference is only in the first round. Player I with Preflop = [1 8 ] knows already that

Player II has Preflop [ 42 46 ] but he is only uncertain about which of the 2 possible sets of the River

applies. For him the probabilities are equal. The same holds for player I with Preflop [ 42 46]. Similar

considerations apply to Player II. If we would have chosen a random sample then, with a very high

probability, there would not have been such a pattern at all: the optimal player has exactly the same

information as the fictive player, making detailed analysis worthless.

The following table 1.2.1.2 gives the card names and the coded poker values at the river.The coded

poker value contains 5 numbers: nr 1 gives the main value (1-9) in decreasing value with

9 = SF (straight flush), 8 = 4K (four of a kind), 7 = FH (full house), 6 = F (flush),

5 = S (straight), 4 = 3K (three of a kind), 3 = 2P (two pairs), 2 = 1P (one pair).

1 = 0P ( no pair = high card),

and nrs 2-6 the other meaningful values in decreasing importance (0=no meaning). So, in the first

card combination at a possible showdown player I has a high card with kicker 14 = Ace and player II

has two pairs 13 = King, 12 = Queen with kicker 14 = Ace.

CardComb Player Preflop River Poker value

1

2

3

4

I

II

I

II

I

II

I

II

2c 3s

Qd Kd

2c 3s

Qd Kd

Qd Kd

2c 3s

Qd Kd

2c 3s

8d Ts Qh Ks Ac

2h 2s 3c Tc Jd

8d Ts Qh Ks Ac

2h 2s 3c Tc Jd

1 14 13 12 10 8

3 13 12 14 0 0

7 2 3 0 0 0

2 2 13 12 11 0

3 13 12 14 0 0

1 14 13 12 10 8

2 2 13 12 11 0

7 2 3 0 0 0

Table 1.2.1.2 Corresponding card names and poker values for table 1.2.1.

1.2.2 Representative sample

14

The discussion of the properties of the elementary sample in section 1.2.1 makes clear that we will

not use a random sample. The results for the beginner and fictive player are always unbiased but the

sample must be chosen large for a small standard deviation. The result for the optimal player is

always positively biased towards that of the fictive player. We need a (too) large random sample to

be sure that the optimal player has not much more knowledge compared with the whole population.

It is much better to choose another type of still a representative sample in order to reduce the bias.

The basic idea is already contained in the elementary sample: let the chosen card combinations be

such that for a specific player a preflop has a lot of other preflops of his opponents and also a lot of

rivers. Of course, the selection algoritm must be such that theoretically we get the whole population

by taking the sample sufficiently large.

In table 1.2.2.1 we give an example of the result of a construction of a representative sample in a

specific way.

At first we draw from the deck of 52 cards randomly SimRiver = 2 sets of 5 cards for the River (the

maximum would be MaxSimRiver = C(52,5) = 2 598 960). For each River there remain 47 cards in the

deck. Choose with such a River randomly SimCards = 4 (the minimum, the maximum would be

MaxSimCards = 47). From those 4 cards form all possible SimPlayers = 6 Preflop-combinations of

both players (MaxSimPlayers = C(47,2) x C(45,2) = 1 070 190). So the total numer of card

combinations is TotCombs = 2 x 6 = 12 (MaxTotCombs = 2 781 381 002 400). The final step for the

list is to randomize the order (not shown here).

15


1

2

3

4

5

6

7

8

9

10

11

12

30 35

30 38

30 43

35 38

35 43

38 43

9 23

9 27

9 46

23 27

46 9

27 46

38 43

35 43

35 38

30 43

30 38

30 35

27 46

23 46

23 27

9 46

9 27

9 23

1 3 10 47 48

1 3 10 47 48

1 3 10 47 48

1 3 10 47 48

1 3 10 47 48

1 3 10 47 48

21 24 36 38 45

21 24 36 38 45

21 24 36 38 45

21 24 36 38 45

21 24 36 38 45

21 24 36 38 45

Table 1.2.2.1 CardCombs with Players =2, MaxRounds =2, SimCards =4, SimRiver =2.

Of course the sample of table 1.2.2.1 is much to small. For moderate accuracy we take SimRiver = 20,

SimCards = 30 (=> SimPlayers = 164 430).This gives TotCombs =

3 288 600. A sketch of such a representative sample is given in table 1.2.2.2. We use this sample for

the whole analysis of games with MaxRounds = 2 and Players = 2. For this sake the sample has been

saved in a file.


1

2

..

3288600

37 50

7 46

.. ..

6 37

16 38

15 33

.. ..

8 26

10 11 24 29 36

10 11 24 29 36

.. .. .. .. ..

15 19 21 31 49

Table 1.2.2.2 Card numbers of representative sample of 3 288 600 card combinations.

The same procedure for getting representative samples can be followed in other cases. For example,

for a game with 3 rounds (Preflop, Flop, River) and 2 players the table 1.2.2.3 gives a (much too

small) representative sample with MaxRounds = 2, Players = 2, SimFlop = 2 and SimRiver = 2

(combined with each flop). Again we show the list without randomization.

CardComb Preflop I Preflop II Flop River

1

2

3

4

5

6

7

8

9

10

11

12

13

14

25 33 51 52 5 10 46 12 24

25 51 33 52 5 10 46 12 24

25 52 33 51 5 10 46 12 24

33 51 25 52 5 10 46 12 24

33 52 25 51 5 10 46 12 24

51 52 25 33 5 10 46 12 24

2 12 39 50 5 10 46 18 37

2 39 12 50 5 10 46 18 37

2 50 12 39 5 10 46 18 37

12 39 2 50 5 10 46 18 37

12 50 2 39 5 10 46 18 37

39 50 2 12 5 10 46 18 37

11 24 44 46 21 38 39 2 33

11 44 24 46 21 38 39 2 33

16

15

16

17

18

19

20

21

22

23

24

11 46 24 44 21 38 39 2 33

24 44 11 46 21 38 39 2 33

24 46 11 44 21 38 39 2 33

44 46 11 24 21 38 39 2 33

1 5 28 31 21 38 39 25 52

1 28 5 31 21 38 39 25 52

1 31 5 28 21 38 39 25 52

5 28 1 31 21 38 39 25 52

5 31 1 28 21 38 39 25 52

28 31 1 5 21 38 39 25 52

Table 1.2.2.3 CardCombs with MaxRounds= 3, SimCards=4, SimRiver=2, SimFlop =2.

1.3 Skill

For determining the (relative) skill we only have to specify te beginner strategies. From those the

optimal and fictive strategies follow, leading to the skill.

We assume that a beginner will base his decision on direct available card information while

comparing the pot odds with the probability odds (see [1]). Otherwise stated, at any stage of the

game he is able to calculate his expected gain based on his own cards and the available community

cards. In fact, this is a really ‘advanced’ beginner because that is only possible after studying some

poker literature. This seems to be the only way to get a simple formulation of beginner strategies

suitable for all poker variants. Of course, this will lead to an underestimate of skill. This fits in the

policy to search for poker variants with low skill. However, this should be kept in mind when judging

specific skill values.

We consider 4 types of beginner strategies: take the decision with

- maximal expected gain (section 1.3.1), a rather passive strategy,

- randomized decisions as variation (section 1.3.2) leading to bluffing and sandbagging,

- implied pot odds as variation (section 1.3.3),

- maximal bet under those decisions with positive max.exp.gain (section 1.3.4), a rather

aggressive strategy.

We conclude this section with a discussion about the accuracy of the results (section 1.3.5).

1.3.1 Maximal expected gain

17

1.3.1.1 Calculations for the elementary sample

Table 1.3.1.1 gives for both players the win- and equal probabilities at a possible showdown for the

card combinations of table 1.2.1.1 (or table 1.2.1.2):

CardComb Player Preflop

Win Equal

River

Win Equal

1

2

3

4

I

II

I

II

I

II

I

II

0.2924 0.0613

0.6241 0.0198

0.2924 0.0613

0.6241 0.0198

0.6241 0.0198

0.2924 0.0613

0.6241 0.0198

0.2924 0.0613

0 0.2030

0.7828 0.0040

0.9848 0.0020

0.3879 0.0091

0.7828 0.0040

0 0.2030

0.7828 0.0040

0.9848 0.0020

Table 1.3.1.1 Win-and equal probabilities for the players for table 1.2.1.1.

These probabilities have been obtained by enumarating all possible card combinations. They have

been saved in files and form the basis for the calculation of the beginner strategies.

Beginners:

We assume that beginners base their decisions in any decision node on the expected gain given the

pot size, their own preflop and the common river cards in the second round. In poker terminoligy:

they base their decisions by comparing the pot odds with the (probability) odds.

In this section we assume that they take always the decision with maximal expected gain, supposing

that they can actually calculate the desired probabilities accurately enough.We give the calculation

details for the beginner decisions for CardComb=1 in table 1.3.1.2. Its contents will be clear with

figure 1.1.1. E.g., for Node =1 the pot is [2 4] (PotVec with the SB and BB)) with sum 6. Player I is the

current player CurPlayer who has to decide between 3 decisions leading to the next nodes 2, 3, 4

(NextNodes) by Fold with bet 0, Call with bet 2 and Raise with bet 6. The corresponding expected

gains (ExpGainVec) can be calculated from this. E.g., for the decision Call we get the expectation (see

table 1.3.1.1):

[0.2924 + 0.0613 / 2) x (6 + 2)] – 2 = 0.5843.

The gain for Fold is 0 and the gain for Raise is -2.1236. So Call is the decision with the maximal

expected gain. This is represented as the randomized decision [0 1 0] (THM0DecsCell).

18

Node = 1 PotVec: [2 4] CurPlayer: 1

NextNodes: {[2 3 4]} (F=0 Ca=2 R=6)

ExpGainVec = 0 0.5843 -2.1236

THM0DecsCell{Node} = 0 1 0


NextNodes: {[7 8]} (Ch=0 B=4)

ExpGainVec = 5.0720 3.6080

THM0DecsCell{Node} = 1 0


NextNodes: {[5 6]} (F=0 Ca=4)

ExpGainVec = 0 6.1441



NextNodes: {[9 10]} (F=0 Ca=2)




NextNodes: {[12 13]} (Ch=0 B=8)

ExpGainVec = 12.5576 10.8364



NextNodes: {[14 15]} (Ch=0 B=8)

ExpGainVec = 1.6242 -5.5636



NextNodes: {[16 17 18]} (F=0 Ca=8 R=16)

ExpGainVec = 0 -4.7515 -11.9394

THM0DecsCell{Node} =1 0 0


NextNodes: {[21 22 23]} (F=0 Ca=8 R=16)

ExpGainVec = 0 17.1152 15.3939


Node = 18 PotVec: [24 16] CurPlayer:2

NextNodes: {[19 20]} (F=0 Ca=8)




NextNodes: {[24 25]} (F=0 Ca=8)

ExpGainVec = 0 -3.1273



NextNodes: {[27 28]} (Ch=0 B=8)

ExpGainVec = 6.2788 4.5576



NextNodes: {[29 30]} (Ch=0 B=8)

ExpGainVec = 0.8121 -6.3758



NextNodes: {[31 32 33]} (F=0 Ca=8 R=16)

ExpGainVec = 0 -5.5636 -12.7515



NextNodes: {[36 37 38]} (F=0 Ca=8 R=16)

ExpGainVec = 0 10.8364 9.1152

THM0DecsCell{Node} =0 1 0


NextNodes: {[34 35]} (F=0 Ca=8)




NextNodes: {[39 40]} (F=0 Ca=8)




NextNodes: {[42 43]} ( Ch=0 B=8)

ExpGainVec = 12.5576 10.8364



NextNodes: {[44 45]} (Ch=0 B=8)

ExpGainVec = 1.6242 -5.5636



NextNodes: {[46 47 48]} (F=0 Ca=8 R=16)

ExpGainVec = 0 -4.7515 -11.9394



NextNodes: {[51 52 53]} (F=0 Ca=8 R=16)

ExpGainVec = 0 17.1152 15.3939



NextNodes: {[49 50]} (F=0 Ca=8)




NextNodes: {[54 55]} (F=0 Ca=8)



Table 1.3.1.2 Calculation of beginners decisions for CardComb = 1

The calculations for CardComb = 2, 3, 4 are the same: see table 1.3.1.3. Ordered by the player nodes,

this leads to table 1.3.1.3.

19

Player I

Node CardComb = 1 CardComb = 2 CardComb = 3 CardComb = 4

1

8

12

13

23

27

28

38

42

43

53

0 1 0

0 1

1 0

1 0 0

1 0

1 0

1 0 0

1 0

1 0

1 0 0

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

0 1 0

0 1

1 0

1 0 0

0 1

1 0

1 0 0

0 1

1 0

1 0 0

0 1

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

Player II


3

4

11

15

18

26

30

33

41

45

48

1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1

1 0

1 0 0

1 0

1 0

1 0 0

1 0

1 0

1 0 0

1 0

1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

1 0

0 1 0

0 1

Table 1.3.1.3 Beginner strategies in randomized form for maximal exp. gain.

Note that for nodes in round = 1 the decisions for CardComb 1 and 2 have to be the same since the

preflop is the same. The same holds for CardComb 3 and 4.

Substitution of the beginner strategies of table 1.3.1.3 leads to the result of table 1.3.1.4:

CardComb EndNode Result Player I

Gain Bet

Player II

Gain Bet

1

2

3

4

29

29

29

29

II wins SD

I wins SD

I wins SD

II wins SD

-4 4

4 4

4 4

-4 4

4 4

-4 4

-4 4

4 4

Sum 0 16 0 16

Exp. 0 4 0 4

Table 1.3.1.4 Gains and total bets for all beginners.

In fact this result is rather trivial for decisions based on maximal expected gain. Let p be the

probability of winning the showdown, S the pot size before the decision and B the bet amount.Then

Check is always better than Bet since

E(Check) = pS > E(Bet)= p(S + B) – B.

Furthermore, let C be the call amount and R the raise amount (R > C). Then Call is always better than

Raise since

20

E(Call) = p (S + C) – C > E(Raise)= p(S + R) – R.

Finally, at node 1 Call is always better than Fold regardless of the cards. The minimal win probability

for a preflop is p = 0.2924 (for [2c 3d] ). So in this case

E(Call) = p (6 +2 ) – 2 = 0.3392 > E(Fold) = 0.

Therefore any game with beginners develops as 1 => 3=> 7 => 26 => 27 => 29.

Fictive players:

An (optimal) fictive player knows the cards of his opponent and the river cards before the betting

starts. Together with the knowlegde of the beginner strategy of his opponent the maximal gain can

be calculated for each CardComb. The first step is table 1.3.1.5, also needed for the optimal player.

We comment the result for CardComb = 1 given in table 1.3.1.5. At first consider the fictive or

optimal player I against the beginner II. Place arrows in figure 1.1.1 in the leaves of the decisions of

beginner II according to table 1.3.1.3. This determines 5 possible endnodes of the fictive or optimal

player I (EndNodeVec). From this and table 1.2.1.1 for the showdown the 5 corresponding gains and

bets follow (GainEndVec and BetEndVec). The results for the fictive or optimal player II follows in the

same way. For other CardCombs and tables of the beginner the results follow also in the same way.

CardComb OptFictPlayer I OptFictPlayer II

1 EndNodeVec

GainEndVec

BetEndVec

2 14 22 29 37

-2 -8 -16 -4 -12

2 8 16 4 12

29 31 44 46

4 4 8 8

4 12 8 16

2 EndNodeVec

GainEndVec

BetEndVec

2 14 22 29 37

-2 8 16 4 12

2 8 16 4 12

29 32 44 47

-4 -12 -8 -16

4 12 8 16

3 EndNodeVec

GainEndVec

BetEndVec

2 14 21 29 36

-2 8 8 4 4

2 8 16 4 12

29 32 44 47

-4 -12 -8 -16

4 12 8 16

4 EndNodeVec

GainEndVec

BetEndVec

2 14 22 29 37

-2 -8 -16 -4 -12

2 8 16 4 12

29 32 44 47

4 12 8 16

4 12 8 16

Table 1.3.1.5 Basic calculations for optimal and fictive players.

Now consider the fictive players. For CardComb = 1 and FictPlayer = I take the maximal gain (FictGain

= -2) with the corresponding Bet (FictBet = 2). This gives the first results in Table 1.3.1.6. For the

other CardCombs and FictPlayer = II the results follow in the same way.

CardComb OptFictPlayer I

Gain Bet

OptFictPlayer II

Gain Bet

1

2

3

4

-2 2

16 16

8 8

-2 2

8 8

-4 4

-4 4

16 16

21

Sum 20 28 16 32

Exp. 5 7 4 8

Table 1.3.1.6 Gains and bets for the fictive players.

Optimal players:

The set of all card distributions is the elementary sample of table 1.2.1.1. This small sample makes

that the knowledge of the optimal player is almost the same as that of the fictive player. The

difference is only in the first round. Player I with Preflop = [1 8 ] knows already that Player II has

Preflop [ 42 46 ] but he is only uncertain about which of the 2 possible sets of the River applies. For

him the probabilities are equal.

The same holds for player I with Preflop [ 42 46]. Similar considerations apply to Player II. We

describe in detail the calculation of the expected gains and bet for the optimal Player I against the

beginner II. The explained framework is also suitable for more complex cases.

Calculations are backwards in the game tree. We start with the endnode results for the optimal and

fictive player I as given in the table 1.3.1.5 and do the calculations backwards. It follows

automatically that betting rounds are also taken backwards. In a specific round we take the maximal

expected gain over the possible decisions for the optimal player I; for the other beginners we fill in

their decisions given in table 1.3.1.4. If we go back one round then we have to sum up the results

with the same information.

The following table with subsequent indices for different card combinations based on table 1.2.1.1 is

helpful. So we see for player I that for CardComb = 1 and = 2 the indeces are the same for the Preflop

(OptInx= 1) but different for the River.

Player I Player II

CardComb Preflop River

1 1 1

2 1 2

3 2 3

4 2 4

CardComb Preflop River

1 1 1

2 1 2

3 2 3

4 2 4

Table 1.3.1.7 The indices OptInx for CardCombs of the players.

Table 1.3.1.8 gives for the optimal player I the backward calculations. Substitute the beginner

strategy of player II in the game tree. Nodes which cannot be reached get a bet equal to 0. The first

node that can be reached is node 30 of player II. Calculate the expected gains (OptGain) and bets

(OptBet) for all different card combinations (OptInx). For this pure strategy this means simply

copying.

The next node is 27 of the optimal player I. Take the maximal expected gain over the decisions for all

different card combinations (OptGain) and denote the corresponding bet (OptBet) and the number

22

of the decision (OptDec). For equal expected gains we take the one with the lowest bet. Since

optimal strategies can always be taken pure a randomized form is not necessary.

So we continue up till the last node 11 for which round = 2 (River).

23

LastNode = 55, MaxRInx = 4

Round = 2 Node = 30 (Player = 2)

OptInx NextOptGainVec OptGain NextOptBetVec OptBet

1

2

3

4

0 -12 0

0 12 0

4 0 0

0 -12 0

-12

12

4

-12

0 12 0

0 12 0

12 0 0

0 12 0

12

12

12

12


OptInx NextOptGainVec OptGain NextOptBetVec OptBet OptDec

1

2

3

4

-4 -12

4 12

4 4

-4 -12

-4

12

4

-4

4 12

4 12

4 12

4 12

12

12

12

12

1

2

1

1



1

2

3

4

-4 0

12 0

4 0

-4 0

-4

12

4

-4

4 0

12 0

4 0

4 0

4

4

4

4



1

2

3

4

0 -16 0

0 16 0

16 0 0

0 -16 0

-16

-16

16

-16

0 16 0

0 16 0

16 0 0

0 16 0

16

16

16

16


OptInx NextOptGainVec OptGain NextOptBetVec OptBet OpDec

1

2

3

4

-8 -16

8 16

8 8

-8 -16

-8

16

8

-8

8 16

8 16

8 16

8 16

8

16

8

8

1

2

1

1



1

2

3

4

-8 0

16 0

8 0

-8 0

-8

16

8

-8

8 0

8 0

8 0

8 0

8

16

8

8

Table 1.3.1.8 The calculations for Round = 2 of optimal player I.

Before we get to round 1 (Preflop) we have to sum up the results with the same information on the

base of table 1.3.1.7. Hence, for node 11 we sum the results for OptInx = 1 and 2 and put it in table

1.3.1.9 in node 6. The same has to be done for nodes 26 and 41.

LastNode = 10 (NextRound), MaxRInx = 2

OptInx 1 2 3 4 5 11

=>

6

26

=>

7

8 9 41

=>

10

1 Gain

Bet

0

0

-4

4

0

0

0

0

0

0

8

24

8

16

0

0

0

0

0

0

2 Gain

Bet

0

0

4

4

0

0

0

0

0

0

0

16

0

8

0

0

0

0

0

0

Table 1.3.1.9 The transformation of the results to Round = 1 of optimal player I

24

With table 1.3.1.9 we continue as before with round = 1. The result is shown in table 1.3.1.10. The

last step is always node 1.

25



1

2

0 8

0 16

8

16

0 24

0 16

24

16



1

2

8 0

0 0

8

0

16 0

8 0

16

8


OptInx NextOptGainVec OptGain NextOptBetVec OptBet OpDec

1

2

-4 8 8

-4 0 0

8

0

4 16 24

4 8 16

16

8

2

2

Sum 8 24

Exp. 2 6

Table 1.3.1.10 The calculations for Round = 1 of optimal player I.

Summation in tabel 1.3.1.10 gives SumOptGain = 8 and SumOptBet = 24. Since all 4 CardCombs have

equal probability this leads for the optimal Player I to OptGain = 8 / 4 = 2 and OptBet = 24 / 4 = 6.

For the optimal player II the calculations are the same.

Table 1.3.1.11 gives the (pure) optimal optimal strategies of both players. Of course, for nodes that

cannot be reached due to the beginner strategy of the opponent, the choice is arbitrary and has been

filled with 1 for the smallest bet. For comparison we have included the beginner strategies of table

1.3.1.3 as well.

Player I

Node OptInx = 1 OptInx = 2 OptInx = 3 OptInx = 4

Beginner Opt Beginner Opt Beginner Opt Beginner Opt

1

8

12

13

23

27

28

38

42

43

53

0 1 0 2

0 1 1

1 0 1

1 0 0 1

1 0 1

1 0 1

1 0 0 1

1 0 1

1 0 1

1 0 0 1

1 0 1

0 1 0 2

0 1 1

1 0 2

0 1 0 1

0 1 1

1 0 2

0 1 0 1

0 1 1

1 0 1

0 1 0 1

0 1 1

1 0 1

1 0 0 1

0 1 1

1 0 1

1 0 0 1

0 1 1

1 0 1

1 0 0 1

0 1 1

1 0 1

0 1 0 1

0 1 1

1 0 1

0 1 0 1

0 1 1

1 0 1

0 1 0 1

0 1 1

Player II

Node OptInx = 1 OptInx = 2 OptInx = 3 OptInx = 4

Beginner Opt Beginner Opt Beginner Opt Beginner Opt

3

4

11

15

18

26

30

1 0 1

0 1 1

1 0 1

0 1 0 1

0 1 1

1 0 1

0 1 0 1

1 0 1

0 1 1

1 0 1

0 1 0 1

0 1 1

1 0 1

0 1 0 1

1 0 1

1 0 0 1

1 0 1

1 0 1

1 0 0 1

1 0 1

0 1 0 1

0 1 1

1 0 2

0 1 0 1

26

33

41

45

48

0 1 1

1 0 1

0 1 0 1

0 1 1

0 1 1

1 0 1

0 1 0 1

0 1 1

1 0 1

1 0 1

1 0 0 1

1 0 1

0 1 1

1 0 2

0 1 0 1

0 1 1

Table 1.3.1.11 Beginner and optimal strategies for maximal exp. gain

The next table 1.3.1.12 gives an overview of the gains and bets for both players for the different

types beginner, optimal and fictive player.

Gain

I II

Bet

I II

Beginner 0 0 4 4

Optimal 2 2 6 6

Fictive 5 4 7 8

Table 1.3.1.12 The expected gains for all types of players.

Skill:

If we measure the game result simply as (expected) gain, then by taking the mean (or sum) of the

gains over the 2 players we get from table 1.3.1.11:

LE = 2, RE = 2 ½ => S = 4/9 = 0.4444.

This is the way the calculation of skill is explained in [1].

Of course there are other possibilities. We make a distiction in notation. Since the foregoing measure

is based on the gain and the sum of both players we use the notation Sgs ( = S). We can also take the

mean of the skill based on gain of the two players, leading to Sgm. Another possibility is to consider

the return rate as the game result. This leads to measures Srs and Srm. Table 1.3.1.13 gives an

overview of the calculations based on table 1.3.1.12.

EGainArr EBetArr Rarr=

I II gmean | I II mean rmean | I II

0 0 0 | 4 4 4 | 0 | 0 0

2 2 2 | 6 6 6 | 2/6 | 1/3 1/3

5 4 9/2 | 7 8 15/2 | 9/15 | 5/7 1/2

--- --- ---- ----- ---- ----

Sm 2/5 1/2 Sgs Srs | Sr 7/15 2/3

--------- -----------

Sgm Srm

=> Sgs Srs Sgm Srm

4/9 5/9 9/20 17/30

0.4444 0.5556 0.4500 0.5667

27

Table 1.3.1.13 Different forms of skill measures for maximal expectations.

If the order of magnitude of the bets is more or less the same we think that the measure Sgs is the

most simple one, and therefore the most appropriate. The 4 measures can differ substantially only

for special constructed theoretical games. In practical examples the difference is always small. For

signalyzing irregularities we compute always the 4 values and refer to S = Sgs as the relative skill.

1.3.1.2 Results for the representative sample

Table 1.3.1.14 gives the results for the representative sample (table 1.2.2.2). It has been put in a

general form, suitable for all variants to come later on.

At first the game parameters are shown. We take always SB = 2 and BB = 4. Pot-limit and no-limit are

disabled (values 0): see sections 7 and 8, respectively. BetVec = 4 means that the only size of the bet

is 4, the usual value for fixed-limit: see section 6 for spread-limit. DoubleRound = 2 is the first round

in which all bets are doubled. The starting chips of the players (ChipVec = [100 100]) have been

chosen in such a way that they are sufficient for both players and any decision in the game.

Then the beginner parameters follow. GainLabel = 1 means maximal expected gains (this section),

the meaning of the other value GainLabel = 0 is explained in section 1.3.4. Implied odds are disabled

for both players (FactDecVec = [ 0 0 ]): see section 1.3.2. Randomization is also disabled for both

players (RandProbDecVec = [0 0 ] ): see section 1.3.3.

The game parameters together with the beginner parameters generate the gametree: TotNodes = 55

(see figure 1.1.1 or appendix A3).

Thereafter the information about sampling follows. The data come from the file as sketched in table

1.2.2.2. The meaning of the sim-parameters is discussed already there.

The bias-test gives insight in the bias of the results of the optimal players. Their gains are calculated

for 10 intermediate sample sizes. So, based on the first part TotCombVec = 328 860 (of the whole

sample size of TotCards = 3 288 600) the gains of the optimal players are [1.6218 1.5937] (under

MGainArr) and the corresponding bets [8.9975 9.1165] (under MbetArr). We see that the gains

decrease with increasing sample sizes, indicating a decreasing positive bias. At the end this bias has

almost vanished; fluctuations are only due to dispersion (see section 1.3.4).

Then the gains and bets of all players are given and the skill values follow. In particular, S = Sgs =

0.3105, larger than the upper juridical threshold 0.3.

Finally, the computer time for the calculations is given: 37228.322697 seconds = 0.4309 days.

28

Game parameters:

Players = 2

Small Blind = 2 - Big Blind = 4

PotLimit = 0 - NoLimit = 0

BetVec = 4

MaxRaises = 1 - MaxRounds = 2 - DoubleRound = 2

RaiseVec = [ ]

ChipVec = 100 100

Beginner parameters:

GainLabel = 1

FactDecVec = 0 0

RandProbDecVec = 0 0

TotNodes = 55

Data from file:

SimCards = 30 MaxSimCards = 47

SimPlayers = 164430 MaxSimPlayers = 1070190

SimRiver = 20 MaxSimRiver = 2598960

TotCombs = 3288600

Bias test:

TotCombVec MGainArr MBetArr

P1 P2 P1 P2

328860 1.6218 1.5937 8.9975 9.1165

657720 1.6088 1.5634 9.0747 9.1160

986580 1.5923 1.5620 9.0565 9.0893

1315440 1.5859 1.5581 9.0269 9.1284

1644300 1.5855 1.5517 9.0426 9.1047

1973160 1.5842 1.5481 9.0532 9.0591

2302020 1.5811 1.5495 9.0443 9.0937

2630880 1.5798 1.5492 9.0695 9.0999

2959740 1.5739 1.5525 9.0721 9.1040

3288600 1.5715 1.5527 9.0735 9.1016

ExpGainArr:

P1 P2

beg 0 0

opt 1.5715 1.5527

fict 5.4615 4.5999

ExpBetArr:

beg 4.0285 4.0285

opt 9.0735 9.1016

fict 7.9411 8.7314

Sgs,Srs,Sgm,Srm:

0.3105 0.2848 0.3126 0.2878

Elapsed time is 37228.322697 seconds = 0.4309 days.

Table 1.3.1.14 Results for maximal expectations (MaxRaise = 1).

1.3.2 Randomized decisions

29

The beginner strategies of table 1.3.1.3 can be modified from pure to mixed strategies. In poker

terminology: we incorperate sandbagging and bluffing. The main decision gets probability 0.9 and the

remaining probability 0.1 is equally distributed among the other decicions as far as they differ from

the decision Fold. We take these values for both players (RandProbDecVec = [0.1 0.1]. This leads to

the modified beginner strategies of table 1.3.2.1.

Player I


1

8

12

13

23

27

28

38

42

43

53

0 0.9 0.1

0 1

0.9 0.1

0.9 0.05 0.05

0.9 0.1

0.9 0.1

0.9 0.05 0.05

0.9 0.1

0.9 0.1

0.9 0.05 0.05

0.9 0.1

0 0.9 0.1

0 1

1 0

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0 0.9 0.1

0 1

1 0

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0 0.9 0.1

0 1

1 0

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

Player II


3

4

11

15

18

26

30

33

41

45

48

0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 1

0.9 0.1

0.9 0.05 0.05

0.9 0.1

0.9 0.1

0.9 0.05 0.05

0.9 0.1

0.9 0.1

0.9 0.05 0.05

0.9 0.1

0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

0.9 0.1

0 0.9 0.1

0 1

Table 1.3.2.1 Beginner strategies for randomization.

In case of randomized decisions for beginners the calculation of the gain for the fictive player can be

quite cumbersome, even if we do this in the most appropriate way.

We give as example the calculation of the expected gain for CardComb = 1 and Fictive Player = 2. At

first we sketch why things become more complicated than for pure strategies (see table 1.3.1.5 and

1.3.1.6) . As the first step we take the tree of figure 1.1.1 and fill in at the leaves the (conditional)

probabilities of the randomized strategy of the beginner. For each leave of the fictive player we

substitute the (conditional) probability 1, indicating that this node can be chosen by him. From that

we calculate the absolute probabilities for the endnodes. In the table 1.3.2.2 we restrict ourselves to

endnodes with positive probability.

Node Prob Gain Bet Node Prob Gain Bet

14 0.09 8 8 36 0.09 -4 4

30

16 0.09 8 16

17 0.005 16 16

19 0.005 -16 16

20 0.005 24 24

21 0.01 -8 8

22 0.01 16 16

24 0.009 16 24

25 0.001 24 24

29 0.81 4 4

31 0.81 4 12

32 0.045 12 12

34 0.045 -12 12

35 0.045 20 20

37 0.09 12 12

39 0.081 12 20

40 0.009 20 20

44 0.81 8 8

46 0.81 8 16

47 0.045 16 16

49 0.045 -16 16

50 0.045 24 24

51 0.09 -8 8

52 0.09 16 16

54 0.081 16 24

55 0.009 24 24

Table 1.3.2.2 Abs.probs. of the first step for fictive player II with CardComb = 1

Note that for pure strategies all nodes would have prob 1. We take the one with maximal gain:

node = 20, maxgain = 24, maxbet = 24, abs.prob. = 0.005.

For a pure strategy of the beginner we get always abs.prob.= 1 and we are ready. However, now in

only a fraction 0.005 (FreeProb) of the cases this maximal gain is attainable. Therefore we have to

decide what is optimal in the remaining cases with fraction 0.995 (ProbFict). We continue now in the

same way by recalculating the conditional probabilities given that node 20 is not chosen. This leads

to another chosen node. We repeat this with all chosen nodes untill there are no remaining cases any

longer (ProbFict = 0). We describe for this particular case the whole calculation process precisely in

the form of an iterative algoritm.

We start formally with FictGain = 0, FictBet = 0, ProbFict = 1 and the conditional probs of table

1.3.2.1. For the fictive player II we substitute for all cond.probs 1, indicating that all leaves can be

chosen.

Step 1:

We determine the set of nodes with positive abs. probs. We find that endnode 20 has maximal gain

MaxGain = 24 with corresponding bet MaxBet = 24.

Then we go one node backwards from this endnode 20. This gives node 13 with the 3rd leave. This

node has the cond.probs [0.9 0.05 0.05]. The 3rd element gives the current free probability:

FreeProb = 0.05 and not-normalized new vector [0.9 0.05 0]. Normalization to sum 1 gives the new

cond.probs [0.9474 0.0526 0].

Then we continue with the next backnodes with node 1 as the last one.

From node 13 we go backwards with leave 2 to node 11. This node belongs to the optimal player II,

so we do not change the ones as this leaves can still be chosen.

From node 11 we go backwards with leave 2 to node 6 of beginner I. This node has only one leave, so

nothing can change: FreeProb = 0.05 and cond.probs [1].

31

From node 6 we go backwards with leave 2 to node 4 of the optimal player II, so nothing changes.

From node 4 we go backwards with leave 3 to node 1 of the beginner I with cond.probs [0 0.9 0.1].

Then the current free prob becomes FreeProb = FreeProb x 0.1 = 0.05 x 0.1 = 0.005, leading to [0 0.9

0.1-0.005] = [0 0.9 0.0995] and after normalization to [0 0.9045 0.0955], the new cond.probs.

Finally, we update FreeProb and ProbFict:

FreeProb = FreeProb x ProbFict = 0.005 x 1 = 0.005,

ProbFict = ProbFict – FreeProb = 1 – 0.005 = 0.995,

and the gain and bet:

FictGain = FictGain + MaxGain x FreeProb = 0.12

FictBet = FictBet + MaxBet x FreeProb = 0.12.

Step 2:

With the new ProbFict and the new cond.probs we continue as under step 1.

Last Step:

The iteration stops at the step with ProbFict = 0. Then FictGain is the maximal gain of the fictive

player II. Table 1.3.2.3 shows all intermediate steps. In this particular case we have MaxBet =

MaxGain for all steps, implying the same for FictBet and FictGain. Therefore we have omitted the

bet-columns.

32

Step FreeProb Node MaxGain FictGain ProbFict

1 0.0050 20 24 0.12 0.9950

2 0.00095 25 24 0.1428 0.9940

3 0.0450 50 24 1.2228 0.9490

4 0.0086 55 24 1.4280 0.9405

5 0.0423 35 20 2.2745 0.8982

6 0.0080 40 20 2.4353 0.8901

7 0.0050 17 16 2.5145 0.8852

8 0.0081 22 16 2.6441 0.8771

9 0.0419 47 16 3.3145 0.8352

10 0.0686 52 16 4.4115 0.7666

11 0.0361 32 16 4.8445 0.7305

12 0.0590 37 12 5.5531 0.5715

13 0.0810 14 8 6.2011 0.5905

14 0.5905 44 8 10.9250 0

Table 1.3.2.3 Successive steps for FictGain for fictive player II with CardComb = 1.

The calculation of the expected gains of beginners and optimal players is done in the same way as in

section 1.3.1: randomized decisions of beginners give no new problems. So, this leads to the

expected gains and bets for all types of players for all CardCombs. Taking the means over the 4

CardCombs gives table 1.3.2.4.

Gain

I II

Bet

I II

Beginner 0.0172 -0.0172 6.2168 6.2330

Optimal 2.8700 2.5000 9.0700 10.5000

Fictive 6.3058 5.2498 8.3058 9.2498

Table 1.3.2.4 The expected gains for all types of players for randomization.

Finally, this leads to table 1.3.2.5 of the different forms of skill measures.

Sgs Srs Sgm Srm

0.4647 0.4169 0.4658 0.4185

Table 1.3.2.5 The expected gains for all types of players for randomization.


Table 1.3.2.6 gives the results for the representative sample (table 1.2.2.2).

33

Game parameters:

Players = 2 - WithBB = 1



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDec = 0 0

RandProbDecVec = 0.1000 0.1000

TotNodes = 55

Data from file:




TotCombs = 3288600

Bias test:


328860 1.9522 1.7177 10.0645 9.8385

657720 1.9378 1.6847 10.1193 9.8089

986580 1.9199 1.6832 10.0850 9.8096

1315440 1.9134 1.6797 10.1001 9.8214

1644300 1.9132 1.6725 10.1273 9.7866

1973160 1.9112 1.6689 10.1430 9.7835

2302020 1.9078 1.6704 10.1118 9.7931

2630880 1.9068 1.6702 10.1320 9.7939

2959740 1.9007 1.6740 10.1060 9.8125

3288600 1.8979 1.6744 10.1066 9.7968

ExpGainArr:

0.0126 -0.0126

1.8979 1.6744

6.6880 5.7785

ExpBetArr:

6.1819 6.2048

10.1066 9.7968

9.1647 9.9290

Sgs,Srs,Sgm,Srm:

0.2865 0.2749 0.2869 0.2757


Table 1.3.2.6 Results for randomized decisions (MaxRaise = 1).

1.3.3 Implied odds

The beginner strategies of table 1.3.1.3 can be modified in such a way that possible future gains are

added to the pot before expectations are taken. In poker terminoligy: we incorperate implied pot

odds. More specific, at each decision node for a specific player and a specific decision he takes the

product of the corresponding decision chips DecChips and the number CntDecisions of players with

34

bet decisions in the same round. He multiplies this number of chips with an optimistic factor FactDec

and adds this to the real pot. This gives a fictive pot. Then he takes his decision based on the maximal

expectations of this fictive pot. The factor FactDec may depend on the player: the higher the more

optimistic. In the special case that FactDec = 0 for all players, we get back table 1.3.1.3.

We work out the example that FactDec = 1 for both players. In table 1.3.3.1 we give the chips that

will be added to the pot dependent on the decisions to be taken. Only positive amounts are shown.

For example, in node 1 Player I has CntDecisions = 2 and DecChips = 2 for the possible decision Call

(product 4), and for the Raise decision CntDecision = 1 and DecChips = 6 (product 6). For all other

nodes in the table we have CntDecisions = 1.


Ca=4 R=6


B=4


B=8


B=8


R=16


R=16


B=8


B=8


R=16


R=16


B=8


B=8


R=16


R=16

Table 1.3.3.1 Fictive addional amounts to the pot for FactDec = 1.

Now expectations can be calculated exactly in the same way as it is done in table 1.3.1.2 and the

chosen decisions follow: table 1.3.3.2. Comparing with table 1.3.1.3 we see that now often decisions

with more chips are taken.

Player I


1

8

12

13

23

27

28

38

42

43

53

0 1 0

0 1

1 0

1 0 0

1 0

1 0

1 0 0

1 0

1 0

1 0 0

1 0

0 1 0

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1 0

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1 0

0 1

1 0

0 0 1

0 1

1 0

0 0 1

0 1

1 0

0 0 1

0 1

Player II


3

4

11

15

18

26

30

0 1

0 1

0 1

0 0 1

0 1

1 0

0 0 1

0 1

0 1

1 0

0 0 1

0 1

1 0

0 0 1

1 0

0 1

1 0

1 0 0

1 0

1 0

1 0 0

1 0

0 1

0 1

0 0 1

0 1

0 1

0 0 1

35

33

41

45

48

0 1

0 1

0 0 1

0 1

0 1

1 0

0 0 1

0 1

1 0

1 0

1 0 0

1 0

0 1

0 1

0 0 1

0 1

Table 1.3.3.2 Beginner strategies for implied odds.

With this starting point we can calculate the gains and bets of the optimal and fictive players and

thereafter the skill: see table 1.3.3.3.

36

Gain

I II

Bet

I II

Beginner 0 0 16 16

Optimal 4 4 10 8

Fictive 7 7 9 10

Sgs Srs Sgm Srm

0.5714 0.6032 0.5714 0.6143

Table 1.3.3.3 The expected gains, bets and skill for future gains


Table 1.3.3.4 gives the results for this sample (table 1.2.2.2; GainLabel = 1).

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 1 1


TotNodes = 55

Data from file:




TotCombs = 3288600

Bias test:


328860 3.1152 2.4870 12.7424 11.2571

657720 3.0740 2.4257 12.8287 11.2473

986580 3.0419 2.4162 12.8766 11.2829

1315440 3.0277 2.4064 12.8965 11.2770

1644300 3.0282 2.3963 12.9139 11.2693

1973160 3.0234 2.3899 12.9108 11.2551

2302020 3.0138 2.3918 12.9192 11.2697

2630880 3.0104 2.3902 12.9227 11.2559

2959740 3.0040 2.3936 12.9235 11.2722

3288600 2.9999 2.3941 12.9317 11.2587

ExpGainArr:

-0.2741 0.2741

2.9999 2.3941

6.9779 6.2097

37

ExpBetArr:

13.8833 13.8687

12.9317 11.2587

9.5126 10.5484

Sgs,Srs,Sgm,Srm:

0.4090 0.3392 0.4043 0.3366


Table 1.3.3.4 Results for implied odds (MaxRaise = 1).

1.3.4 Maximal bet under positive expectations

In section 1.3.1 beginners choose for the decisions with maximal expectations. Another more

aggressive way is to choose under all possible decisions with positive expectations that one with the

maximal bet. So, if Call and Raise have both positive expectations then the beginner chooses Raise.

The strategies follow easily with table 1.3.1.3:

Player I


1

8

12

13

23

27

28

38

42

43

53

0 1 0

0 1

1 0

1 0 0

1 0

1 0

1 0 0

1 0

1 0

1 0 0

1 0

0 1 0

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

1 0

0 1 0

0 1

0 1

0 1 0

0 1

Player II


3

4

11

15

18

26

30

33

41

45

48

0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 1

0 1

0 1 0

0 1

1 0

0 1 0

0 1

0 1

0 1 0

0 1

1 0

0 1

1 0

1 0 0

1 0

1 0

1 0 0

1 0

1 0

1 0 0

1 0

1 0

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

0 1

0 0 1

0 1

Table 1.3.4.1 Beginner strategies for maximal bet under positive exp. gain.

Gain

I II

Bet

I II

Beginner 2 -2 16 16

Optimal 4 4 10 10

Fictive 7 6 9 10

Sgs Srs Sgm Srm

38

0.6154 0.5846 0.5750 0.5727

Table 1.3.4.2 The expected gains, bets and skill for maximal bet


Table 1.3.4.3 gives the results for the representative sample of table 1.2.2.2 (GainLabel = 0).

39

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 0

FactDecVec = 0 0


TotNodes = 55

Data from file:




TotCombs = 3288600

Bias test:


328860 2.3978 2.1658 13.3749 11.9443

657720 2.3702 2.0520 13.5788 11.9027

986580 2.3394 2.0319 13.5778 11.9248

1315440 2.3272 2.0177 13.6121 11.9424

1644300 2.3262 2.0023 13.6186 11.9436

1973160 2.3227 1.9939 13.5810 11.9502

2302020 2.3150 1.9938 13.5850 11.9669

2630880 2.3125 1.9914 13.5893 11.9588

2959740 2.3042 1.9945 13.5742 11.9660

3288600 2.2997 1.9939 13.5670 11.9760

ExpGainArr:

0.2796 -0.2796

2.2997 1.9939

6.8752 6.0921

ExpBetArr:

16.6369 16.6346

13.5670 11.9760

9.4241 10.4961

Sgs,Srs,Sgm,Srm:

0.3311 0.2582 0.3315 0.2606


Table 1.3.4.3 Results for maximal bet under positive expected gains (MaxRaise = 1).

1.3.5 Standard deviations

In the foregoing sections 1.3.1 – 1.3.4 attention has been paid to the bias in the results of the optimal

player. The representative sample has been chosen so large that this bias is small. In this section it

will appear that this holds also for the standard deviation in the skill. We give details for the case

with the lowest skill: randomized decisions with skill S = 0.2865 (see sections 1.3.2.2).

40

Table 1.3.5.1 gives the result of 8 repetitions, of which the first one is table 1.3.2.6.

41

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 55




TotCombs = 3288600

Sample 1:

ExpGainArr:

0.0126 -0.0126

1.8979 1.6744

6.6880 5.7785

ExpBetArr:

6.1819 6.2048

10.1066 9.7968

9.1647 9.9290

Sgs,Srs,Sgm,Srm:

0.2865 0.2749 0.2869 0.2757

Sample 2:

ExpGainArr:

0.0136 -0.0136

1.9649 1.7127

6.6768 5.7568

ExpBetArr:

6.1961 6.2189

10.0226 9.7787

9.1024 9.8540

Sgs,Srs,Sgm,Srm:

0.2958 0.2832 0.2960 0.2837

Sample 3:

ExpGainArr:

0.0118 -0.0118

1.9831 1.7476

6.6453 5.7433

ExpBetArr:

6.1610 6.1840

9.9492 9.7153

9.1467 9.8994

Sgs,Srs,Sgm,Srm:

0.3011 0.2917 0.3014 0.2924

Sample 4:

ExpGainArr:

0.0126 -0.0126

1.8711 1.6452

6.7413 5.8218

ExpBetArr:

6.1628 6.1860

10.0132 9.7721

9.1246 9.9078

Sgs,Srs,Sgm,Srm:

0.2799 0.2692 0.2802 0.2699

Sample 5:

ExpGainArr:

0.0119 -0.0119

2.0414 1.7990

6.7336 5.8029

ExpBetArr:

6.1787 6.1997

10.0709 9.8736

9.1311 9.8935

Sgs,Srs,Sgm,Srm:

0.3063 0.2922 0.3067 0.2929

Sample 6:

ExpGainArr:

0.0123 -0.0123

1.9068 1.6549

6.6258 5.7195

ExpBetArr:

6.1618 6.1851

9.8512 9.7074

9.1811 9.9209

Sgs,Srs,Sgm,Srm:

0.2885 0.2818 0.2887 0.2822

Sample 7:

ExpGainArr:

0.0132 -0.0132

2.0202 1.7840

6.7148 5.7959

ExpBetArr:

6.1944 6.2168

10.1203 9.9432

9.1731 9.9266

Sgs,Srs,Sgm,Srm:

0.3041 0.2895 0.3044 0.2902

Sample 8:

ExpGainArr:

0.0137 -0.0137

1.9617 1.7300

6.7879 5.8734

ExpBetArr:

6.2106 6.2337

10.2491 9.9688

9.2098 9.9863

Sgs,Srs,Sgm,Srm:

0.2916 0.2768 0.2919 0.2776

Table 1.3.5.1 Result of 8 repetitions for the case of randomized decisions (MaxRaise = 1).

42

From the 8 samples in table 1.3.5.1 we get the following mean and (estimated) standarddeviation

(table 1.3.5.2):

Sgs Srs Sgm Srm

Mean = 0.2942 0.2824 0.2945 0.2831

Std = 0.0092 0.0084 0.0093 0.0084

Table 1.3.5.2 Means and standard deviations of 8 repetitions.

Taking accuracy = 2 x standarddeviation, we roughly get a range of 0.02 . This is enough for the

conclusions that will be based on it. The same holds for the simulation results for other skill measures

and beginner strategies.

1.3.6 Overview

Table 1.3.6.1 gives an overview of the results in sections 1.3.1 – 1.3.4. According to section 1.3.5 the

deviation in S can be at most 0.02.

Beginner strategies Table S=Sgs

Maximal exp.gain 1.3.1.14 0.31

Randomized decisions 1.3.2.6 0.29

Implied odds 1.3.3.4 0.41

Maximal bet 1.3.4.3 0.43

Table 1.3.6.1 Overview for skill (MaxRounds =2, Players = 2).

We repeat that the results are all based on the same sample of card distribution (table 1.2.2.2). The

strategy: maximal expectation with a little bit randomization gives the lowest skill. Implied odds and

maximal bet give a significant higher skill.

We are interested in comparisons of the skill with the juridical bound. Therefore in the next sections

with other game parameters we will only investigate the skill for beginner strategies of maximal

expected gain with randomization. We take as randomization the value 0.1. Other values lower or

higher increase the skill (results omitted).

1.4 Going all-in

In the foregoing sections the number of chips to start with was 100 for both players: ChipVec = [100

100]. This is large enough to take every desired decision. In fact this could have been reduced to 24

without changing figure 1.1.1. But if at a certain stage the number of chips is less than (or equal to) to

43

the chips needed for a decision, a player can still take this decision with all his chips. Then his

remaining chips are 0 and further decisions of him are not allowed: he is going all in. The rule is that a

player with no chips cannot take decisions further on. He takes part in the showdown only for the

total chips he has put into the pot. So we have to consider the influence of going all-in. The remaining

players continue to play for the remainder of the pot, the so called side pot. For Players = 2 this is

rather simple: if one player goes all in, the game ends at last when it would be his turn again.

However, for Players >=3 the other players can continue. This can become quite complicated and

more than one side pot can occur.

In any case, the game tree depends on the number of chips if the total maximal bet of decisions is

larger than the number of starting chips of a player.

In this section we illustrate this effect with the full game tree of figure 1.1.1 for the case that ChipVec

= [25 7]. So player I starts with 25 chips and II with 7. Figure 1.4.1 gives the reduction of the game

tree of figure 1.1.1.

Figure 1.4.1 The game tree for all-in with ChipVec = [ 25 7].

At node 3 player II bets all-in with the 3 remaining chips. Then at node 8 player I makes the last

decision. At nodes 4 and 15 player II calls all-in with the 3 remaining chips and has only 7 chips

44

contributed to the pot. So the sidepot for the showdown is [ 7 7]. The remaining pot is simply for

Player I. Table 1.4.2 gives for different ChipVecs the total nodes of the resulting game tree.

I | II 25 23 15 7 3 1

25

23

15

7

3

1

55 49 40 19 3 3

49 43 40 19 3 3

40 40 37 19 3 3 16 61 16 16 3 3

3 3 3 3 3 3

1 1 1 1 1 1

Table 1.4.2 Total nodes dependent on ChipVec.

Note that for Player I starting with 1 chip the game is reduced to pure gambling. Of course, such

games are never played in fixed-limit THM-cash games but can appear in a specific game in THM-

tournements.

1.5 Strategies from a game-theoretic perspective

In the foregoing section 1.3 no attention has been paid to the optimal and fictive strategies

themselves because only the corresponding expected gains and bets are needed for determining the

skill. But optimal play against a given (e.g. beginner) strategy is an interesting topic itself. Even more

interesting is the best play against unknown strategies of the opponents. This is a typical topic in

game theory. It has nothing to do with skill and so this section can be omitted if one is not interested

in strategies for optimal play.

We say that a strategy profile of the set of all players form a (Nash-)equilibrium if for each particular

player his strategy is optimal given the strategies of the other players. Every game has at least one

equilibrium. Clearly, if for a certain game the beginner strategies together form an equilibrium then

the skill of this game is 0. But this has only theoretical value because for practical games (e.g. full

THM, bridge and chess) equilibria cannot even be calculated and therefore will never be played.

For two-person zero-games a little bit more can be said (e.g. THM without rake). By definition, the

gain of player I, say, is always the loss of player II. An equilibrium consists of a so called maximin-

strategy of player I (maximizing his minimal expected gain taken over the strategies of player II), and

a minimax-strategy of player II (minimizing his maximal loss token over the strategies of player I). The

corresponding expected gain of I (loss of II) is the same for each equilibrium and is called the value of

the game.

In this section we can say a little bit about the value and corresponding strategies of the simplified

fixed-limit THM-version of section 1: the base game. The usual way to calculate minimax-strategies of

45

a game is to transform it to its normal form and use linear programming. But this gives only success

for very simple games, certainly not for the THM-basicgame. However, another possibility is to find

the solution by a modified form of fictitious play straightforward from the game tree (the extensive

form of the game). This iteration technique can be applied by extending the computer program for

skill in such a way that it gives also the optimal strategy of a player against given strategies of his

opponents. There remains one serious objection: the convergence of the iteration procedure is

extremely slow.

Table 1.5.1 gives for THM-basic the result of fictitious play. We start with the beginner strategies of

section 1.3.1.2: simply maximal expected gain. The iteration process has been stopped after TotRpts

= 482 iterations (about 250 days).

46

NoOptFictPl = 1 - Rpts = 24

Players = 2 - WithBB = 1



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100

GainLabel = 0

FactDec = 0 0


TotNodes = 55

Data from file:




TotCombs = 3288600

TotRpts = 1

ExpGainArr:

-0.3516 0.3516

1.8244 5.0168

ExpBetArr:

14.2835 13.9554

12.6737 19.4443

TotRpts = 2

ExpGainArr:

-0.6129 0.6129

2.1988 3.0855

ExpBetArr:

14.3756 15.0318

12.7908 18.5625

................................................................

TotRpts = 481

ExpGainArr:

0.5207 -0.5207

0.5697 -0.4718

ExpBetArr:

10.7187 10.1288

10.5151 9.9656

TotRpts = 482

ExpGainArr:

0.5207 -0.5207

0.5696 -0.4719

ExpBetArr:

10.7182 10.1283

10.5219 9.9647

Table 1.5.1 Fictitious play after 482 rounds

Although there is hardly a difference between TotRpts = 481 and TotRpts = 482, there is still not a

sharp convergence. For that we need the equality between the ‘beginners’ and the ‘optimal’ players:

for player I the difference is still 0.5696 - 0.5207 = 0.0489, and for player II 0.0488. This prevents

47

sharp conclusions but gives more than only a qualitative description. A rough estimate for the

minimax-value of the game is + 0.52. This means that player I has the advantage over player II. Of

course this is due to the fact that the SB player I has to invest less in the pot than the BB player II.

Convergence of the corresponding optimal maximin- and minimax strategies of the players has been

far from reached. Table 1.5.2 gives a rough impression on the base of equities. The Eq(uity) of a card

combination is the expected gain of the pot of size 1, in this case Eq = WProb + EProb/2 with WProb

the winning probability and EProb the equal probability. Decisions for card combinations with more

ore less the same Eq have been averaged. In the table equities and probs in strategies are in

percentages (i.e. multiplied by 100).

Node 1: Eq Strat

30 33 2 65

40 32 2 66

50 12 2 86

60 7 1 91

70 6 1 93

80 0 1 99

90 0 1 99

Node 3: Eq Strat

30 67 33

40 56 44

50 40 60

60 23 77

70 13 87

80 0 100

90 0 100

Node 4: Eq Strat

30 31 69

40 32 68

50 12 88

60 7 93

70 5 95

80 0 100

90 0 100

Node 8: Eq Strat

30 31 69

40 32 68

50 14 86

60 8 92

70 7 93

80 0 100

90 0 100

Node: 11

Eq Strat

0 51 49

10 56 44

20 80 20

30 91 9

40 100 0

50 99 1

60 94 6

70 78 22

80 48 52

90 42 58

100 45 55

Node: 12

Eq Strat

0 2 98

10 11 89

20 68 32

30 87 13

40 99 1

50 97 3

60 83 17

70 36 64

80 8 92

90 0 100

100 0 100

Node: 13

Eq Strat

0 96 0 4

10 96 0 4

20 86 9 5

30 61 28 11

40 43 51 6

50 35 58 7

60 31 65 5

70 7 92 2

80 1 85 14

90 0 25 75

100 0 0 100

Node: 15

Eq Strat

0 100 0 0

10 99 0 1

20 94 4 2

30 71 24 6

40 45 51 4

50 33 63 4

60 29 68 3

70 4 94 2

80 0 82 17

90 0 15 85

100 0 0 100

Node: 18

Eq Strat

0 100 0

10 100 0

20 99 1

30 83 17

40 89 11

50 80 20

60 64 36

70 48 52

80 31 69

90 3 97

100 0 100

Node: 23

Eq Strat

0 100 0

10 100 0

20 100 0

30 87 13

40 96 4

50 86 14

60 73 27

70 50 50

80 24 76

90 2 98

100 0 100

Node: 26

Eq Strat

0 95 5

10 97 3

20 99 1

30 99 1

40 100 0

50 99 1

Node: 27

Eq Strat

0 6 94

10 33 67

20 87 13

30 80 20

40 99 1

50 91 9

Node: 28

Eq Strat

0 98 0 2

10 96 1 3

20 85 10 5

30 78 13 9

40 66 27 7

50 45 50 5

48

60 98 2

70 95 5

80 89 11

90 84 16

100 84 16

60 76 24

70 31 69

80 5 95

90 0 100

100 0 100

60 33 64 3

70 4 91 5

80 1 73 27

90 0 25 75

100 0 0 100

Node: 30

Eq Strat

0 100 0 0

10 99 0 1

20 90 9 2

30 76 20 4

40 61 37 2

50 41 56 3

60 32 65 2

70 4 91 4

80 1 72 27

90 0 14 86

100 0 0 100

Node: 33

Eq Strat

0 100 0

10 100 0

20 99 1

30 91 9

40 88 12

50 79 21

60 68 32

70 52 48

80 26 74

90 1 99

100 0 100

Node: 38

Eq Strat

0 100 0

10 100 0

20 99 1

30 86 14

40 95 5

50 87 13

60 74 26

70 48 52

80 24 76

90 2 98

100 0 100

Node: 41

Eq Strat

0 5 95

10 18 82

20 68 32

30 85 15

40 99 1

50 98 2

60 89 11

70 68 32

80 40 60

90 31 69

100 38 62

Node: 42

Eq Strat

0 43 57

10 52 48

20 82 18

30 86 14

40 99 1

50 97 3

60 81 19

70 37 63

80 7 93

90 0 100

100 0 100

Node: 43

Eq Strat

0 98 0 2

10 98 0 2

20 87 9 4

30 59 31 9

40 42 52 6

50 32 62 6

60 28 68 5

70 5 93 2

80 0 85 14

90 0 22 78

100 0 0 100

Node: 45

Eq Strat

0 99 0 1

10 98 0 2

20 86 11 3

30 67 27 6

40 48 48 4

50 40 56 4

60 32 65 3

70 5 92 3

80 0 81 18

90 0 16 84

100 0 0 100

Node: 48

Eq Strat

0 100 0

10 100 0

20 99 1

30 90 10

40 86 14

50 79 21

60 64 36

70 47 53

80 30 70

90 3 97

100 0 100

Node: 53

Eq Strat

0 100 0

10 100 0

20 100 0

30 89 11

40 96 4

50 85 15

60 69 31

70 41 59

80 21 79

90 2 98

100 0 100

Table 1.5.2 Approximate optimal (maximin- and minimax) strategies.

So, the preflop 2c3d gives the minimal equity 0.3203, or rounded off 30%. In the table we see that in

node 1 the decision of player I is [ 33 2 65], randomization with 33% Fold, 2% Call and 65%

Raise.The preflop AcAd gives the maximal equity 0.8520, or rounded of 90%.

The strategy profile of table 1.5.2 can be applied in practice if we can make a rough guess of the

winning probability of the card combination. The effect of the equal probability is almost always

negligible. In the table it is interesting to observe the role of bluffing and sandbagging reflected in the

decision probabilities.

49

2. GENERAL FIXED-LIMIT THM

2.1 Introduction

In section 1 we have analyzed the THM-basic game: a simplified form of fixed-limit THM with the

simplifying parameters Players = 2, MaxRounds = 2, MaxRaises = 1. It appeared that beginner

strategies with randomized decisions give the lowest skill. Therefore we continue only with this

choice and analyze what will happen with the skill for larger values of these parameters. The rules

that are used for full THM are Players >=2, MaxRounds = 4 and MaxRaises =3. The difficulty with

these parameters is that the game tree becomes very large. Table 2.1.1 gives a specification.

MaxRounds Pl = 2 Pl = 3 Pl = 4 Pl = 5 ...

2 MaxRaise = 1

MaxRaise = 2

MaxRaise = 3

3 MaxRaise = 1

MaxRaise = 2

MaxRaise = 3

4 MaxRaise = 1

MaxRaise = 2

MaxRaise = 3

55 509 3902 27307 ...

121 2579 41927 ...

211 11171 ...

280 5634 86287 ...

856 59322 ...

1912 ...

1405 ... ...

6001 ...

17221 ...

Table 2.1.1 Number of nodes for different values of parameters.

Furthermore, for MaxRounds > 2 a representative sample must be taken in a different way to

emphasize the increasing information the optimal player has in each round. This makes that the

needed computer time to do the required calculations becomes too large. Therefore we vary these

parameters independently of each other and analyze the change in skill. In section 2.2 we do this for

MaxRaises, in section 2.3 for Players and in section 2.4 for MaxRounds. In section 2.5 we will

extrapolate to the effect of changing these parameters simultaneously.

So starting point in the analysis is section 1 for Players = 2, MaxRounds = 2, MaxRaises = 1. For the

beginner strategies we use the randomization as explained in section 1.3.2.

2.2 More raises

2.2.1 MaxRaises = 2

50

We change MaxRaises = 1 (table 1.3.2.6) to MaxRaises = 2. Figure 2.2.1.1 gives a part of the game

tree: only the first round. The whole game tree counts 121 nodes (see table 1.2.1). Note that the bet

at node 3 is counted as a raise. So at node 14 there is no choice for another raise.

Figure 2.2.1.1 Gametree for MaxRaises = 2, MaxRounds = 2 and Players = 2.

2.2.2 MaxRaises = 3

Table 2.2.1.1 gives the result if we change MaxRaises = 1 (table 1.3.2.6) to MaxRaises = 3, the number

used in practice.

51

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 211

Data from file:




TotCombs = 3288600

Bias test:


328860 2.2280 2.0248 10.8533 10.8160

657720 2.2112 1.9875 10.9248 10.7803

986580 2.1913 1.9855 10.8920 10.7883

1315440 2.1841 1.9812 10.8809 10.7989

1644300 2.1837 1.9729 10.9260 10.7723

1973160 2.1817 1.9689 10.9520 10.7474

2302020 2.1777 1.9706 10.9207 10.7710

2630880 2.1768 1.9703 10.9378 10.7668

2959740 2.1700 1.9747 10.9096 10.7720

3288600 2.1669 1.9750 10.9333 10.7812

ExpGainArr:

0.0127 -0.0127

2.1669 1.9750

7.9633 7.0967

ExpBetArr:

6.2798 6.3027

10.9333 10.7812

10.4492 11.2473

Sgs,Srs,Sgm,Srm:

0.2750 0.2748 0.2753 0.2753


Table 2.2.1.1 Result for randomized decisions (MaxRaises = 3).

Comparing the results with MaxRaises = 1 we see that there is not much difference between the two

settings. Our conclusion is that the number of allowed raises in THM does not change the skill

significantly. However, there is one important practical difference: the needed computer time for

MaxRaises = 3 is 5 x as large as for MaxRaises = 1.

2.3 More players

52

At first we consider the generalization of the simplified fixed-limit THM-game from 2 players (section

1.1) to 3 players. Note that in this case player I has the SB, II the BB and that III starts the real betting.

Assuming that their starting chips are large enough to avoide going all-in, the game tree grows from

55 to 509 nodes. This is too large to make a complete sketch like figure 1.1.1. In the figures 2.3.1 we

have only displayed the first betting round.

53

Figure 2.3.1 Main Figure 2.3.1a (Node 2)

Figure 2.3.1b (Node 3) Figure 2.3.1c (Node 4)

54

Player I has the SB, player II the BB and player III starts the real betting. Therefore in the second

round player I starts again the betting unless he has already folded (e.g. node 56).

As beginner strategies for the 3 players we consider only the case of randomized decisions of section

1.3.2 for all of them. The calculation of expectations is based on winning probabilities. More

specifically, for a specific player we need now 3 probabilities: the highest card value (Win = 1), equal

card values with one other player (Win = 2) and all equal cardvalues (Win = 3). Enumerating and

saving all possible card combinations takes too much computer time and space. So, as an

approximation we calculate the probs under the assumption that the preflops of all opponents are

drawn with replacement given the cards of the specific player and other open cards. Then we can

calculate the desired probablities PW from P2 of 2 players, which were saved in files (section

1.3.1.1). The general formula is (P = Players, W = Win):

PW(W) = C(P-1,W-1) x P2(2)^(W-1)... x P2(1)^(P-W), 1 <= W <= P.

Take the following example:

P2(1) = 0.2924 (win), P2(2)= 0.0613 (equal).

Then for 3 players we get:

PW(1) = 0.0855, PW(2) = 0.0358, PW(3) = 0.0038

and for 4 players:

PW(1) = 0.0250, PW(2) = 0.0157, PW(3) = 0.0033, PW(4) = 0.0002 .

With these approximations of probabilities we can calculate gain expectations and proceed as in

section 1.3 to obtain the beginner strategies. In particular this holds for randomized decisions as in

section 1.3.2.

2.3.1 Elementary sample

We start with an extension of the elementary sample for 2 players of section 1.2.1: see table 2.3.1.1.

Again this is just intended to explain the analysis and has no practical value.

CardComb Prefl.I Prefl.II Prefl.III River

1

2

3

4

5

6

7

8

9

10

1 8

1 8

1 8

1 8

42 46

42 46

42 46

42 46

29 40

29 40

42 46

42 46

29 40

29 40

1 8

1 8

29 40

29 40

1 8

1 8

29 40

29 40

42 46

42 46

29 40

29 40

1 8

1 8

42 46

42 46

26 36 43 48 49

3 4 5 33 38

26 36 43 48 49

3 4 5 33 38

26 36 43 48 49

3 4 5 33 38

26 36 43 48 49

3 4 5 33 38

26 36 43 48 49

3 4 5 33 38

55

11

12

29 40

29 40

42 46

42 46

1 8

1 8

26 36 43 48 49

3 4 5 33 38

Table 2.3.1.1 Card numbers of an elementary sample of 12 card combinations.

Table 2.3.1.2 gives the relevant card names and poker values.

Card numbers Card names Poker Values

1 8 26 36 43 48 49

1 8 3 4 5 33 38

42 46 26 36 43 48 49

42 46 3 4 5 33 38

29 40 26 36 43 48 49

29 40 3 4 5 33 38

2c 3s 8d Ts Qh Ks Ac

2c 3s 2h 2s 3c Tc Jd

Qd Kd 8d Ts Qh Ks Ac

Qd Kd 2h 2s 3c Tc Jd

9c Js 8d Ts Qh Ks Ac

9c Js 2h 2s 3c Tc Jd

1 14 13 12 10 8

7 2 3 0 0 0

3 13 12 14 0 0

2 2 13 12 11 0

5 14 0 0 0 0

3 11 2 10 0 0

Table 2.3.1.2 Card names and poker values.

For CardComb = 1 and Player =1 the preflop-winning probs are just the probs of the example in the

foregoing introduction: 0.2924 (Win = 1), 0.0613 (Win = 2). The river-probs are: 0 (Win = 1 ),

0.2030 (Win = 2). From these probs the exected gain can be calculated. This leads for this CardComb

to the best decision (maximal exp. gain). Results for the beginner strategies of the other players and

card combinations follow in the same way. From that we can calculate the gains of the beginners,

optimal players and fictive players. We omit the lenghtly details now. The final result is given in table

2.3.1.2. Of course, since the sample is very small the gain of the optimal player has a high positive

bias and this explains the high unrealistic skill value.

ExpGainArr:

I II III

B 0 1.3333 -1.3333

O 7.3333 7.0000 8.5000

F 9.1667 8.0000 10.6667

ExpBetArr:

B 3.6667 4.0000 1.3333

O 9.3333 10.0000 9.3333

F 9.5000 10.3333 8.6667

Sgs: Srs: Sgm: Srm:

0.8204 0.8156 0.8231 0.8342

Table 2.3.1.2 Results for the elementary sample for 3 players (max.exp.gain).

56

2.3.2 Representative sample

Table 2.3.2.1 gives the results for a representative sample. The parameters SimCards = 30 and

SimRiver = 20 determine the size TotCombs = 1 663 200.

Game parameters:

Players = 3



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100 100


GainLabel = 1

FactDecVec = 0 0 0

RandProbDecVec = 0.1000 0.1000 0.1000

TotNodes = 509




TotCombs = 1663200

Bias test:


166320 2.5358 2.4774 4.2955 8.8241 9.8405 7.4507

332640 2.5303 2.4546 4.2377 8.8259 9.8536 7.4061

498960 2.5110 2.4453 4.2319 8.8427 9.8480 7.3714

665280 2.5170 2.4310 4.2354 8.8090 9.8353 7.4169

831600 2.5149 2.4278 4.2262 8.8306 9.8534 7.3926

997920 2.5202 2.4207 4.2221 8.8272 9.8383 7.3407

1164240 2.5222 2.4198 4.2247 8.8250 9.8186 7.2991

1330560 2.5202 2.4233 4.2209 8.8287 9.8139 7.3609

1496880 2.5234 2.4198 4.2226 8.8322 9.8158 7.3088

1663200 2.5212 2.4192 4.2279 8.8881 9.8319 7.3455

ExpGainArr:

-0.1375 -0.0914 0.2289

2.5212 2.4192 4.2279

6.8373 5.7183 9.4738

ExpBetArr:

6.1112 6.3903 1.2239

8.8881 9.8319 7.3455

8.8652 9.6575 7.5774

Sgs,Srs,Sgm,Srm:

0.4162 0.4167 0.4153 0.3935


Tabel 2.3.2.1 Result for 3 players (MaxRaises = 1, MaxRounds = 2).

We see that the skill for 3 players is substantially larger than that for 2 players (see table 1.3.2.6).

Furthermore, note that the result for beginner player III is positive, due to the fact that he is not

involved in prepaying blinds.

57

2.4 MaxRounds = 4

We consider the general case that the number of rounds is 4: preflop, flop, turn, river and that the

betting is doubled from the turn on. Tabel 2.4.1 gives the result.

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


SecMakeTree = 0.283

TotNodes = 1405




SimFlop = 8 MaxSimFlop = 22100

SimTurn = 3 MaxSimTurn = 49

SecGenCards = 75.730

TotCombs = 589680

Bias test:


58968 6.0775 5.4849 16.8760 16.2886

117936 5.7336 5.2150 16.8186 16.2835

176904 5.6179 5.1063 16.7041 16.2450

235872 5.5684 5.0526 16.8005 16.2469

294840 5.5242 5.0179 16.7995 16.1880

353808 5.5294 4.9761 16.7494 16.1751

412776 5.5282 4.9488 16.7763 16.1392

471744 5.5307 4.9261 16.8360 16.1114

530712 5.5104 4.9288 16.8000 16.1480

589680 5.4993 4.9204 16.7950 16.1705

ExpGainArr:

0.0216 -0.0216

5.4993 4.9204

14.7062 13.8642

ExpBetArr:

8.3733 8.4103

16.7950 16.1705

17.4587 18.2325

Sgs,Srs,Sgm,Srm:

0.3647 0.3949 0.3645 0.3945


Table 2.4.1 Result for 2 players (MaxRaises = 1, MaxRounds = 4).

58

The step from MaxRounds = 2 (table 1.3.2.6) to MaxRounds = 4 leads to a substantial increase of

skill. This should be kept in mind when judging other variations.

2.5 Combining results

Table 2.5.1 gives an overview of the foregoing variations (S = Sgs).

59

Basic example Table 1.3.2.6 S = 0.2865

MaxRaises = 3 Table 2.2.1.1 S = 0.2750

Players = 3 Table 2.3.2.1 S = 0.4162

MaxRounds = 4 Table 2.4.1 S = 0.3647

Table 2.5.1 Individual variations on the basic example.

Note that in this table possible deviations are about 0.02 ( see section 1.3.4). We conclude that the

increase of the number of raises has a negligible influence. Contrarily, the increase of the number of

players and the number of rounds is substantial. The general save conclusion can be that the skill of

full fixed-limit THM is larger than the juridical upper bound 0.3.

3. SPREAD-LIMIT THM

We take a simple example of spread-limit as a variation of the basic example: see figure 3.1. BetVec =

[4 8] means that a player has the choice between a bet of 4 (as in section 1) and bet of 8. RaiseVec

=[6] means a choice between the usual raise of 2 x the bet or an additional amount 6 extra.

60

Figure 3.1 Spread-limit with BetVec=[4 8] and RaiseVec = [8].

Table 3.2 gives the result.

61

Game parameters:

Players = 2



BetVec = 4 8


RaiseVec = 8

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 211

Data from file:




TotCombs = 3288600

Bias test:


328860 4.2353 3.4490 17.7965 15.5967

657720 4.1980 3.3811 17.9027 15.5014

986580 4.1604 3.3745 17.8630 15.5671

1315440 4.1434 3.3635 17.8935 15.5370

1644300 4.1407 3.3521 17.8928 15.5181

1973160 4.1404 3.3439 17.9374 15.4891

2302020 4.1320 3.3467 17.9037 15.4907

2630880 4.1304 3.3458 17.8677 15.5163

2959740 4.1192 3.3512 17.8561 15.4996

3288600 4.1130 3.3514 17.8730 15.5303

ExpGainArr:

0.0206 -0.0206

4.1130 3.3514

16.7779 14.4178

ExpBetArr:

7.5541 7.5419

17.8730 15.5303

20.2158 19.2316

Sgs,Srs,Sgm,Srm:

0.2393 0.2826 0.2389 0.2827


Tabel 3.2 Spread-limit result for 2 players (MaxRaises = 1, MaxRounds = 2).

Compared with the basic example (table 1.3.2.6) we see a decrease of the skill.

The spread in table 3.2 can be described qualitatively as high-low. Table 3.3 gives the result of a more

refined spread. The skill decreases even more.

Game parameters:

Players = 2



BetVec = 4 6 8 10

62


RaiseVec = 8

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 547

Data from file:




TotCombs = 3288600

Bias test:


328860 4.7263 4.1650 19.2616 17.8413

657720 4.6804 4.0732 19.3370 17.7838

986580 4.6384 4.0614 19.1948 17.7907

1315440 4.6200 4.0462 19.2606 17.8227

1644300 4.6147 4.0320 19.3092 17.7854

1973160 4.6140 4.0203 19.3238 17.7380

2302020 4.6042 4.0240 19.2795 17.7087

2630880 4.6022 4.0231 19.2957 17.7016

2959740 4.5893 4.0287 19.2541 17.7035

3288600 4.5824 4.0290 19.2565 17.7498

ExpGainArr:

0.0257 -0.0257

4.5824 4.0290

20.6804 22.1695

ExpBetArr:

7.9803 7.9837

19.2565 17.7498

24.2830 27.3132

Sgs,Srs,Sgm,Srm:

0.2010 0.2802 0.2016 0.2796

Elapsed time is 1947561.203295 seconds = 22.5412 days

Tabel 3.3 Refined spread-limit result for 2 players (MaxRaises = 1, MaxRounds = 2).

Note that the beginner strategies are based on maximal expected gain. This requires that winning

probabilities must be calculated more accurately with increasing spread. This is too much to require

for a beginner. We have no simple substitute for describing beginner strategies for such cases.

Therefore we omit spread limit results in drawing general conclusions.

4. POT-LIMIT THM

We take a simple example of potlimit-THM as a variation of the basic example.

Comparing the gametree for this pot-limit with that of section 3 for spread-limit, we get that the

part of it shown in figure 3.1 is the same as for pot-limit. The only difference is node 5 with leave R =

63

8 (instead of R = 14) followed at node 9 with Ca = 6 (instead of Ca = 12). Further up in the tree the

structure deviates. Table 4.2 gives the result.

64

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 193

Data from file:




TotCombs = 3288600

Bias test:


328860 4.0219 4.0419 17.3508 18.4438

657720 3.9825 3.9534 17.2447 18.3481

986580 3.9496 3.9476 17.2050 18.3240

1315440 3.9344 3.9397 17.2000 18.3837

1644300 3.9314 3.9264 17.2435 18.3592

1973160 3.9265 3.9193 17.2767 18.3031

2302020 3.9187 3.9227 17.2371 18.3423

2630880 3.9136 3.9208 17.2258 18.2786

2959740 3.9012 3.9289 17.2335 18.3228

3288600 3.8953 3.9304 17.2572 18.3245

ExpGainArr:

0.0157 -0.0157

3.8953 3.9304

14.8799 15.3307

ExpBetArr:

6.7199 6.7496

17.2572 18.3245

18.0368 20.3203

Sgs,Srs,Sgm,Srm:

0.2590 0.2792 0.2591 0.2790


Tabel 4.2 Pot-limit result for 2 players (MaxRaises = 1, MaxRounds = 2).

Compared with the basic example (table 1.3.2.6) we see a small decrease of the skill. This decrease is

lower than for the spread-limit cases in section 4. However, the same objections for beginner

strategies apply.

5. NO-LIMIT THM

We take a simple example of potlimit-THM as a variation of the basic example: see figure 5.1. Note

that the game stops immediately when a player goes all in sice there are only 2 players.

65

66

Figure 5.1 No-limit with ChipVec = [100 100] (MaxRaises = 1, MaxRounds = 2).

Table 5.2 gives the result.

67

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 97

Data from file:




TotCombs = 3288600

Bias test:


328860 24.6636 24.5636 51.2680 51.6521

657720 24.4308 24.2429 51.6546 51.5529

986580 24.3080 24.1809 51.4212 51.3944

1315440 24.2942 24.1559 51.1874 51.3353

1644300 24.2802 24.0843 51.3043 51.1924

1973160 24.2800 24.0529 51.1955 51.0645

2302020 24.2455 24.0651 51.1790 50.8304

2630880 24.2468 24.0589 51.0647 50.8448

2959740 24.2061 24.0900 50.9056 50.9111

3288600 24.1887 24.1030 50.9077 50.9228

ExpGainArr:

0.0107 -0.0107

24.1887 24.1030

46.9817 46.1221

ExpBetArr:

22.8776 22.8886

50.9077 50.9228

51.6570 52.2286

Sgs,Srs,Sgm,Srm:

0.5187 0.5292 0.5187 0.5292


Tabel 5.2 No-limit result with ChipVec = [100 100] (Randomization).

Compared with the basic example (table 1.3.2.6) we see a large increase of the skill. It is a surprise

that there is such a big difference in skill with potlimit (table 4.2). The reason is not the

randomization as table 5.3 shows. Here the beginner is just based on maximal expectation.

68

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 97

Data from file:




TotCombs = 3288600

Bias test:


328860 25.6402 25.4974 50.4747 50.6842

657720 25.3989 25.1677 50.8627 50.7435

986580 25.2764 25.1019 50.7210 50.6572

1315440 25.2665 25.0781 50.5023 50.6950

1644300 25.2505 25.0051 50.5472 50.4925

1973160 25.2505 24.9727 50.5232 50.3981

2302020 25.2158 24.9860 50.5033 50.2244

2630880 25.2173 24.9798 50.3738 50.2213

2959740 25.1762 25.0110 50.1466 50.2702

3288600 25.1596 25.0249 50.2157 50.2676

ExpGainArr:

-0.0000 0.0000

25.1596 25.0249

47.0545 46.1928

ExpBetArr:

4.2986 4.2986

50.2157 50.2676

51.1758 52.0375

Sgs,Srs,Sgm,Srm:

0.5382 0.5528 0.5382 0.5529


Tabel 5.3 No-limit result with ChipVec = [100 100] (No randomization)

We can combine PotLimit and Nolimit. This leads to an extension of the gametree. Tabel 5.4 gives the

result.

69

Game parameters:

Players = 2



BetVec = 4


RaiseVec = [ ]

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 283

Data from file:




TotCombs = 3288600

Bias test:


328860 24.8141 24.7936 51.1117 51.6377

657720 24.5785 24.4688 51.4883 51.5297

986580 24.4560 24.4058 51.2905 51.3542

1315440 24.4430 24.3813 51.0622 51.3044

1644300 24.4288 24.3094 51.1661 51.1470

1973160 24.4287 24.2776 51.0888 51.0267

2302020 24.3940 24.2899 51.0815 50.7900

2630880 24.3949 24.2837 50.9259 50.8019

2959740 24.3544 24.3148 50.7459 50.8589

3288600 24.3371 24.3280 50.7797 50.8749

ExpGainArr:

0.0126 -0.0126

24.3371 24.3280

47.4227 46.6709

ExpBetArr:

20.1154 20.1314

50.7797 50.8749

52.1775 53.2952

Sgs,Srs,Sgm,Srm:

0.5172 0.5366 0.5172 0.5367


Tabel 5.4 Combined Pot- and No-limit result (No randomization)

Surprisingly, there is no much difference in skill with NoLimit alone. If we take a more spread pattern

of bets, the result is the same as shown in table 5.5.

70

Game parameters:

Players = 2



BetVec = 4 6 8


RaiseVec = 4 8

ChipVec = 100 100


GainLabel = 1

FactDecVec = 0 0


TotNodes = 1087

Data from file:




TotCombs = 3288600

Bias test:


328860 24.9518 24.9367 51.0909 51.6746

657720 24.7089 24.6099 51.4591 51.5763

986580 24.5862 24.5464 51.2229 51.3894

1315440 24.5730 24.5227 51.0065 51.3408

1644300 24.5582 24.4505 51.0924 51.1795

1973160 24.5568 24.4185 51.0585 51.0601

2302020 24.5222 24.4311 51.0577 50.8241

2630880 24.5228 24.4249 50.9033 50.8322

2959740 24.4821 24.4561 50.7353 50.8809

3288600 24.4652 24.4696 50.7028 50.8958

ExpGainArr:

0.0185 -0.0185

24.4652 24.4696

48.1735 46.9229

ExpBetArr:

15.3142 15.3209

50.7028 50.8958

53.5390 53.3861

Sgs,Srs,Sgm,Srm:

0.5146 0.5416 0.5147 0.5416


Tabel 5.5 Combined Spread, Pot- and No-limit result (randomization)

The general conclusion is that the skill is very high as soon as no-limit is allowed.

6. CONCLUSIONS

Table 6.1 gives an overview of the foregoing game variations in sections 3 - 5 (S = Sgs).

Basic example Table 1.3.2.6 S = 0.2865

71

No-limit Table 5.2 S = 0.5187

Combined Table 5.5 S = 0.5146

Table 6.1 Game variations on the basic example.

In this tables the results for spread-limit (section 3) and pot-limit (section 4) have been omitted. As

already explained there, the required ability for calculating accurately winning probabilities is too

heavy for beginners.

Noe that in table 6.1 all game variations have the same number of raises and rounds: MaxRaises = 1

and MaxRounds = 4. We have seen in section 2.5 that especially the usual number of MaxRounds = 4

and the number of players do increase the skill substantially. Therefore, we conclude that not only

for full fixed-limit but also for other usual full game variations the skill is larger than the upper

juridival bound 0.3.

72

REFERENCES

[1] Texas Hold ‘Em: a game of skill

For other references we refer to [1].

73

APPENDIX: MATLAB-PROGRAM

In this appendix we describe roughly the MATLAB-program with which all results were obtained.

Note that the program only works on a 64-bit Windows- 7 PC with at least an internal memory of 10

Gb and with a 64-bit MATLAB-version of at least version 7.12 (R2011a).

A1. Files of the program THM.

All files have been put on the disc THMDVD. They should be copied all to the same directory.

There are 3 types of files: m-files with the program code, mat-files as part of the program and mat-

files generated by the program. The following overview does not contain generated files.

m-files with program code:

THM the main program, reads all parameters;

calls CalcTHM.

CalcTHM Used by THM;

does the calculations, loads and saves files of the type CardCombArr**;

eventually saves the file THMStratsFile for strategies and displays results;

calls MakeTreeTotTHM, GenTHMCardCombArr, CalcTHMProbs,

CalcTHMOptCardInx, CalcTHMStrats, CalcSkill.

MakeTreeTotTHM Used by CalcTHM;

generates the game tree.

GenTHMCardComArr used by CalcTHM;

generates the card combinations CardCombArr.

CalcTHMProbs used by CalcTHM;

calculates for CardCombArr the probs;

calls GetProbPreflop, GetProbFlop, GetProbTurn, GetProbRiver

internal function InitGlobalVars uses the following mat-files:

RiverCountsFile, NumComm5CardsVecFile,

TurnCountsFile, NumComm4CardsFile,

FlopCountsFile, NumComm3CardsFile,

PreflopCountsFile, NumPreflopVecFile.

GetProbPreflop used by CalcTHMProbs;

calculates probs for the preflop;

calls TransformPreflopComm, Hand2Nr, BinSearch.

GetProbFlop used by CalcTHMProbs;

calculates probs for the flop;


GetProbTurn used by CalcTHMProbs;

calls TransformPreflopComm, Hand2Nr, BinSearch;

calculates probs for the turn.

GetProbRiver used by CalcTHMProbs;

calculates probs for the river;


TransformPreflopComm used by GetProbPreflop, GetProbFlop, GetProbTurn, GetProbRiver;

transforms for suits in preflop.

74

Hand2Nr used by GetProbPreflop, GetProbFlop, GetProbTurn, GetProbRiver;

returns for a hand the corresponding numerical value.

BinSearch used by GetProbPreflop, GetProbFlop, GetProbTurn, GetProbRiver;

the binary search for numerical values in ordered mat-files.

CalcTHMOptCardInx used by CalcTHM;

administrates card combinations and corresponding probs.

CalcTHMStrats used by CalcTHM;

calculates exp. Gains and bets for all players;

eventually loads, modifies and saves the file THMStratsFile for strategies;

calls MakeTHM0Decs, CalcFictGainBet, CalcTHMOptGainBet, GetGainBetDistr.

MakeTHM0Decs used by CalcTHMStrats;

calculates the beginner strategies.

CalcFictGainBet used by CalcTHMStrats;

calculates the gain and bet of the fictive players.

CalcTHMOptGainBet used by CalcTHMStrats;

calculates the gain and bet of the optimal players and eventually the corresponding optimal

strategies.

GetGainBetDistr used by CalcTHMStrats;

calculates the gain and bets of the endnodes;

internal function calls GetValPokerHands.

GetValPokerHands used by GetGainBetDistr;

returns the poker values of hands.

CalcSkill used by CalcTHM;

calculates the 4 measures of skill from gains and bets.

75

A2. Description of the parameters of the program THM.

We will only describe the the various parameters of the program. In section A2.1 we just indicate

how the results of the tables in the foregoing sections can be reproduced. In section A2.2 we discuss

some other settings for more detailed and additional output.

A2.1 Reproducing tables

We will describe how to set the parameters in the program in order to obtain the results as descibed

in the various sections.

The main program is the script file THM.m in which the game parameters get their values. All other

parts are functions (including subfunctions). The main function is the file CalcTHM.m which calls all

other functions and generates the main output.

% THM

% last update 2011-12-21

% Program to calculate for THM

% skill, strategies for beginners and optimal players

% and do fictitious play dependent on parameters.

% The main program only sets parameters,

% calculations and output is made with CalcTHM.

% external m-function:

% CalcTHM (results are displayed by this procedure)

clear, clear global

Rpts=1; % >1 for fictitious play

NoOptFictPl=0; % =2 for no calc. optimal & fictive player

% =1 for no calc. for fictive player

% =0 for all calculations

WithBB=1; % =1 if BB always makes a decision in 1st round

SaveCards=0; % =1 creates CardCombFile** (MaxRounds Players) for rep.use

SaveText=0; % =1 creates the file THM.txt of results

% set game parameters

SB=2;

BB=4;

BetVec=[4]; % fixed bet amounts

Players=2; % 2<=Players<=9

ChipVec=100*ones(1,Players); % length =Players !!!

RaiseVec=[]; % fixed raise amounts additional above MinRaise

MaxRounds=2; % max.number of rounds <=4

% game rules depending on MaxRounds:

% MaxRnd 1 2 3 4

76

% Round

% 1 River=4 Prefl=1 Prefl=1 Prefl=1

% 2 - River=4 Flop=2 Flop=2

% 3 - - River=4 Turn=3

% 4 - - - River=4

% DoubleRound: double Bet and Raise from Round>=DoubleRound>=2

NoLimit=0; % =1 allowed (only AllIn)

PotLimit=0; % =1 allowed

MaxRaises=1; % max.number of raises

% set beginners parameters

RandProb=0.1; %0 0.1 0.2 0.3

FactDec=0; % 0.2 0.4 0.6 0.8

GainLabel=1; % =1 for max exp.gain, =0 for max dec.chips

% can be set separately:

FactDecVec=FactDec*ones(1,Players); % FactDecVec >= 0

RandProbDecVec=RandProb*ones(1,Players); % 0<=RandProbDecVec<=1

% set up simulation if no CardCombFile

% dummy values

SimFlop=2; % dummy value, only used if MaxRounds>=3

SimTurn=2; % dummy value, only used if MaxRounds>=4

switch Players+10*MaxRounds

case 12

SimCards=30; % always used

SimRiver=20; % always used

% TotCombs = 3,288,600

case 22

SimCards=30;

SimRiver=20;

% TotCombs = 3,288,600

case 32

SimCards=25;

SimRiver=5;

SimFlop=10;

% TotCombs = 3,795,000

case 42

SimCards=15;

SimRiver=2;

SimFlop=10;

SimTurn=2;

% TotCombs = 589,680

case 13

SimCards=12; % always used

SimRiver=20; % always used

% TotCombs = 1,663,200

case 23

SimCards=12;

SimRiver=20;

% TotCombs = 1,663,200

case 33

SimCards=10;

SimRiver=5;

SimFlop=10;

% TotCombs = 945,000

case 43

SimCards=10;

SimRiver=2;

SimFlop=10;

SimTurn=2;

77

case 14 % not used

case 24 % not used

case 34 % not used

case 44 % not used

case 15 % not used

case 25

SimCards=10;

SimRiver=10;

% TotCombs = 1,134,000

case 35

SimCards=10;

SimRiver=2;

SimFlop=5;

% TotCombs = 1,134,000

case 45

SimCards=10;

SimRiver=2;

SimFlop=5;

SimTurn=2;

otherwise

disp('error in simulation parameters'),pause

end

CalcTHM(SB,BB,BetVec,ChipVec,PotLimit,NoLimit,RaiseVec,...

MaxRaises,MaxRounds,WithBB,GainLabel,FactDecVec,RandProbDecVec,...

SimCards,SimRiver,SimFlop,SimTurn,...

SaveCards,SaveText,NoOptFictPl,Rpts)

Table A2.1 The main script file THM.m.

If the program THM is run with the script in table A2.1 as it is, then another sample like those in table

1.3.5.1 is taken and analyzed.

Text written to the screen can be saved to the text-file THM.txt if we change SaveText = 0 to

SaveText = 1.

At first we describe the effect if we only change the parameter SaveCards = 0 to 1. Then the program

checks whether there is a file CardCombFile22.mat ( MaxRounds=2, Players = 2) present in the

current directory. If not, then a new sample with SimCards=30 and SimRiver=20 is taken (see case

22) and at the end of the program this sample is saved with this name. However, if already such a file

exists then the sample in it is taken. So this gives the possibilty to use the same sample again and

again.

So, check that CardCombFile22.mat is present and change to SaveCards = 1. Then we get exactly the

result of table 1.3.2.6.

Now, with SaveCards = 1, change RandProb = 0.1 to RandProb = 0. Then we get table 1.3.1.14.

After that, with SaveCards = 1 and RandProb = 0, change FactDec = 0 to FactDec = 1. Then we get

table 1.3.3.4.

78

After that, with SaveCards = 1, RandProb = 0 and FactDec = 0, change GainLabel = 1 to GainLabel = 0.

Then we get table 1.3.4.3.

Finally, with SaveCards = 1, RandProb = 0.1, FactDec = 0 and GainLabel = 1, change MaxRaises = 1 to

MaxRaises = 3. Then we get table 2.2.1.1.

So, as far as MaxRounds = 2 and Players = 2, all results of section 1 can be checked exactly since the

file CardCombFile22 is saved and read again if SaveCards = 1.

For other values of (MaxRounds, Players) than (2, 2) other samples are used. With

CardCombFile23.mat present, table 2.3.2.1 can be reproduced: take Players = 3.

With CardCombFile42.mat we can reproduce table 2.4.1: take MaxRounds = 4.

Tables 3.2, 3.3, 4.2, 5.2 and 5.3 use again CardCombFile22. For all those tables set RandProb=0.1. For

table 3.2 (spread-limit) set BetVec = [4 8] and RaiseVec = [8], for table 4.2 set PotLimit = 1 and for

table 5.2 set NoLimit = 1.

A2.2 Settings for different and additional output

In the foregoing section A2.1 always Rpts = 1, NoOptFictPl = 0 and WithBB = 1. Other settings of

these parameters leads to all kind of options.

Changing WithBB = 1 to WithBB = 0 leads to game trees for the simplifying THM-rule that the BB-

player gets not necessarily a move in the first round if MaxRaises >=1.

With Rpts = 1, changing NoOptFictPl = 0 to NoOptFictPl = 1 has two effects:

- the calculations for the fictive players are skipped,

- the strategies for the optimal players are also calculated: if more decisions lead to the same

expected gain, then the decision with the lowest bet is taken. These optimal strategies

(THMOptStratsCell) together with the beginner strategies (THM0StratsCell) are saved in the file

THMStratsFile. This file can be used for further analysis of strategies.

If THM.m is executed again then beginner strategies are not calculated again but read from this file.

This file contains also a variable TotRpts with unchanged value 1.

With Rpts = 1, changing NoOptFictPl = 0 to NoOptFictPl = 2 has also two effects:

- the calculations for the fictive and optimal players are skipped,

79

- for the beginners not only expected gains and bets are calculated but also the complete

simultaneous prob.distribution of gains and bets of all players.

The most complex option is the setting Rpts > 1 and NoOptFictPl = 1. Then a loop Rpt = 1: Rpts is

executed. In each step TotRpts = TotRpts + 1 and is saved in THMStratsFile. As long as Rpt < Rpts the

beginner strategies and the optimal strategies are mixed to new beginner strategies: at each decision

node the weigthed sum of the beginner- and optimal decision with weights Weight and 1 – Weight

respectively. The new beginner strategy is saved in the file THMStratsFile. The variable Weight is

defined in the function file CalcTHMStrats. It has the value Weight = TotRpts / (1+TotRpts), leading to

a kind of fictitious play in behaviour game form (instead of the usual normal game form). This whole,

somewhat difficult, procedure with a repeatedly saved file makes it possible to interrupt the

execution of THM.m at any time without loosing the essential information: if the program is

restarted it continues with the last step. At the stage where expected gains of the beginners are

equal to those of the optimal players, a Nash-equilibrium is reached. This has been applied in section

1.5

Of course options are created easily. E.g. if at each new step we can give beginners the just

calculated optimal strategies by changing Weight = TotRpts / (1+TotRpts) to Weight = 0.

80

A3. The game tree function for THM.

The procedure TestMakeTreeTotTHM is an important test function which only generates the game

tree. It only calls MakeTreeTotTHM (see the list in A1). The procedure works also with a 32-bit

version of MATLAB. Below we display the result for the tree in figure 1.1.1. All these data are used in

the calculations for skill.

Players = 2 - Small Blind = 2 - Big Blind = 4

PotLimit = 0 - NoLimit = 0 - BetVec = 4


RaiseVec =[ ]

ChipVec = 100 100

Nodes = 55 Node = 1

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [2 4]

ChipVec: [98 96]

CurPlayer: 1

DecType: ' '

DecChips: 0

CntDecisions: 1

EndState: 0

BackNode: 0

NextNodes: {[2 3 4]}

MinRaise: 4

Raises: 0

Round: 1

Node = 2

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [2 4]

ChipVec: [98 96]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 1

NextNodes: {[]}

MinRaise: 4

Raises: 0

Round: 1

Node = 3

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [4 4]

ChipVec: [96 96]

CurPlayer: 2

DecType: 'Call'

DecChips: 2

CntDecisions: 2

EndState: 0

BackNode: 1

NextNodes: {[7 8]}

MinRaise: 4

Raises: 0

Round: 1

Node = 4

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 4]

ChipVec: [92 96]

CurPlayer: 2

DecType: 'Raise'

DecChips: 6

CntDecisions: 1

EndState: 0

BackNode: 1

NextNodes: {[5 6]}

MinRaise: 4

Raises: 1

Round: 1

Node = 5

FoldVec: [0 1]

AllinVec: [0 0]

PlayVec: [1 0]

PotVec: [8 4]

ChipVec: [92 96]

CurPlayer: 1

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 4

NextNodes: {[]}

MinRaise: 4

Raises: 1

Round: 1

Node = 6

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 1

DecType: 'Call'

DecChips: 4

CntDecisions: 0

EndState: 0

BackNode: 4

NextNodes: {[11]}

MinRaise: 4

Raises: 1

Round: 1

Node = 7

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [4 4]

ChipVec: [96 96]

CurPlayer: 1

DecType: 'Check'

DecChips: 0

CntDecisions: 0

EndState: 0

BackNode: 3

NextNodes: {[26]}

MinRaise: 0

Raises: 0

Round: 1

Node = 8

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [4 8]

ChipVec: [96 92]

CurPlayer: 1

DecType: 'Bet'

DecChips: 4

CntDecisions: 1

EndState: 0

BackNode: 3

NextNodes: {[9 10]}

MinRaise: 4

Raises: 0

Round: 1

Node = 9

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [4 8]

ChipVec: [96 92]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 8

NextNodes: {[]}

MinRaise: 4

Raises: 0

Round: 1

Node = 10

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

Node = 11

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

Node = 12

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

81

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 2

DecType: 'Call'

DecChips: 4

CntDecisions: 0

EndState: 0

BackNode: 8

NextNodes: {[41]}

MinRaise: 4

Raises: 0

Round: 1

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 2

DecType: 'NextR'

DecChips: 0

CntDecisions: 2

EndState: 0

BackNode: 6

NextNodes: {[12 13]}

MinRaise: 0

Raises: 0

Round: 2

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 1

DecType: 'Check'

DecChips: 0

CntDecisions: 1

EndState: 0

BackNode: 11


MinRaise: 0

Raises: 0

Round: 2

Node = 13

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 16]

ChipVec: [92 84]

CurPlayer: 1

DecType: 'Bet'

DecChips: 8

CntDecisions: 1

EndState: 0

BackNode: 11

NextNodes: {[16 17 18]}

MinRaise: 8

Raises: 0

Round: 2

Node = 14

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 2

DecType: 'Check'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 12

NextNodes: {[]}

MinRaise: 0

Raises: 0

Round: 2

Node = 15

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 8]

ChipVec: [84 92]

CurPlayer: 2

DecType: 'Bet'

DecChips: 8

CntDecisions: 1

EndState: 0

BackNode: 12

NextNodes: {[21 22 23]}

MinRaise: 8

Raises: 0

Round: 2

Node = 16

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [8 16]

ChipVec: [92 84]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 13

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 17

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 16]

ChipVec: [84 84]

CurPlayer: 2

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 13

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 18

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [24 16]

ChipVec: [76 84]

CurPlayer: 2

DecType: 'Raise'

DecChips: 16

CntDecisions: 1

EndState: 0

BackNode: 13


MinRaise: 8

Raises: 1

Round: 2

Node = 19

FoldVec: [0 1]

AllinVec: [0 0]

PlayVec: [1 0]

PotVec: [24 16]

ChipVec: [76 84]

CurPlayer: 1

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 18

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 20

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [24 24]

ChipVec: [76 76]

CurPlayer: 1

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 18

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 21

FoldVec: [0 1]

AllinVec: [0 0]

PlayVec: [1 0]

PotVec: [16 8]

ChipVec: [84 92]

CurPlayer: 1

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 15

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 22

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 16]

ChipVec: [84 84]

CurPlayer: 1

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 15

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 23

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 24]

ChipVec: [84 76]

CurPlayer: 1

DecType: 'Raise'

DecChips: 16

CntDecisions: 1

EndState: 0

BackNode: 15


MinRaise: 8

Raises: 1

Round: 2

Node = 24

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [16 24]

ChipVec: [84 76]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 23

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

82

Node = 25

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [24 24]

ChipVec: [76 76]

CurPlayer: 2

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 23

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 26

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [4 4]

ChipVec: [96 96]

CurPlayer: 2

DecType: 'NextR'

DecChips: 0

CntDecisions: 2

EndState: 0

BackNode: 7


MinRaise: 0

Raises: 0

Round: 2

Node = 27

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [4 4]

ChipVec: [96 96]

CurPlayer: 1

DecType: 'Check'

DecChips: 0

CntDecisions: 1

EndState: 0

BackNode: 26


MinRaise: 0

Raises: 0

Round: 2

Node = 28

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [4 12]

ChipVec: [96 88]

CurPlayer: 1

DecType: 'Bet'

DecChips: 8

CntDecisions: 1

EndState: 0

BackNode: 26

NextNodes: {[31 32 33]}

MinRaise: 8

Raises: 0

Round: 2

Node = 29

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [4 4]

ChipVec: [96 96]

CurPlayer: 2

DecType: 'Check'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 27

NextNodes: {[]}

MinRaise: 0

Raises: 0

Round: 2

Node = 30

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [12 4]

ChipVec: [88 96]

CurPlayer: 2

DecType: 'Bet'

DecChips: 8

CntDecisions: 1

EndState: 0

BackNode: 27

NextNodes: {[36 37 38]}

MinRaise: 8

Raises: 0

Round: 2

Node = 31

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [4 12]

ChipVec: [96 88]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 28

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 32

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [12 12]

ChipVec: [88 88]

CurPlayer: 2

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 28

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 33

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [20 12]

ChipVec: [80 88]

CurPlayer: 2

DecType: 'Raise'

DecChips: 16

CntDecisions: 1

EndState: 0

BackNode: 28


MinRaise: 8

Raises: 1

Round: 2

Node = 34

FoldVec: [0 1]

AllinVec: [0 0]

PlayVec: [1 0]

PotVec: [20 12]

ChipVec: [80 88]

CurPlayer: 1

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 33

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 35

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [20 20]

ChipVec: [80 80]

CurPlayer: 1

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 33

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 36

FoldVec: [0 1]

AllinVec: [0 0]

PlayVec: [1 0]

PotVec: [12 4]

ChipVec: [88 96]

CurPlayer: 1

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 30

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 37

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [12 12]

ChipVec: [88 88]

CurPlayer: 1

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 30

Node = 38

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [12 20]

ChipVec: [88 80]

CurPlayer: 1

DecType: 'Raise'

DecChips: 16

CntDecisions: 1

EndState: 0

BackNode: 30

Node = 39

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [12 20]

ChipVec: [88 80]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 38

83

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2


MinRaise: 8

Raises: 1

Round: 2

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 40

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [20 20]

ChipVec: [80 80]

CurPlayer: 2

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 38

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 41

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 2

DecType: 'NextR'

DecChips: 0

CntDecisions: 2

EndState: 0

BackNode: 10


MinRaise: 0

Raises: 0

Round: 2

Node = 42

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 1

DecType: 'Check'

DecChips: 0

CntDecisions: 1

EndState: 0

BackNode: 41


MinRaise: 0

Raises: 0

Round: 2

Node = 43

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 16]

ChipVec: [92 84]

CurPlayer: 1

DecType: 'Bet'

DecChips: 8

CntDecisions: 1

EndState: 0

BackNode: 41

NextNodes: {[46 47 48]}

MinRaise: 8

Raises: 0

Round: 2

Node = 44

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [8 8]

ChipVec: [92 92]

CurPlayer: 2

DecType: 'Check'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 42

NextNodes: {[]}

MinRaise: 0

Raises: 0

Round: 2

Node = 45

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 8]

ChipVec: [84 92]

CurPlayer: 2

DecType: 'Bet'

DecChips: 8

CntDecisions: 1

EndState: 0

BackNode: 42

NextNodes: {[51 52 53]}

MinRaise: 8

Raises: 0

Round: 2

Node = 46

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [8 16]

ChipVec: [92 84]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 43

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 47

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 16]

ChipVec: [84 84]

CurPlayer: 2

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 43

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 48

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [24 16]

ChipVec: [76 84]

CurPlayer: 2

DecType: 'Raise'

DecChips: 16

CntDecisions: 1

EndState: 0

BackNode: 43


MinRaise: 8

Raises: 1

Round: 2

Node = 49

FoldVec: [0 1]

AllinVec: [0 0]

PlayVec: [1 0]

PotVec: [24 16]

ChipVec: [76 84]

CurPlayer: 1

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 48

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 50

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [24 24]

ChipVec: [76 76]

CurPlayer: 1

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 48

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 51

FoldVec: [0 1]

AllinVec: [0 0]

PlayVec: [1 0]

PotVec: [16 8]

ChipVec: [84 92]

CurPlayer: 1

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 45

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

Node = 52

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 16]

Node = 53

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [16 24]

Node = 54

FoldVec: [1 0]

AllinVec: [0 0]

PlayVec: [0 1]

PotVec: [16 24]

84

ChipVec: [84 84]

CurPlayer: 1

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 45

NextNodes: {[]}

MinRaise: 8

Raises: 0

Round: 2

ChipVec: [84 76]

CurPlayer: 1

DecType: 'Raise'

DecChips: 16

CntDecisions: 1

EndState: 0

BackNode: 45


MinRaise: 8

Raises: 1

Round: 2

ChipVec: [84 76]

CurPlayer: 2

DecType: 'Fold'

DecChips: 0

CntDecisions: 0

EndState: 1

BackNode: 53

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

Node = 55

FoldVec: [0 0]

AllinVec: [0 0]

PlayVec: [1 1]

PotVec: [24 24]

ChipVec: [76 76]

CurPlayer: 2

DecType: 'Call'

DecChips: 8

CntDecisions: 0

EndState: 1

BackNode: 53

NextNodes: {[]}

MinRaise: 8

Raises: 1

Round: 2

85

Cash and tournament poker: games of skill?

Report commissioned by W.M.C. van den Berg, examining magistrate in charge of criminal matters in

the district of Amsterdam.

Prof. Peter Borm

Prof. Ben van der Genugten

Department of Econometrics and Operations Research

Tilburg University

Tilburg, September 2009

86

Table of contents

1. Introduction.………………………………………………………………………………. 4

2. Conclusions.……………………………………………………………………………..... 6

3. Dutch case law………………………………………………………………………. 7

3.1. State of affairs regarding cash poker…………………………………….… 7

3.2. Management games……………………………………………………………….... 8

3.3. Review and skill threshold level………………………………………………... 9

4. Method of relative skill..................………………………………………………... 9

4.1. General…………………………………………………………………………. . 10

4.2. Single-player games……………………………………………………………... 11

4.3. Multiple-player games…………………………………………………………….... 12

4.4. Theoretical and practical validation ……………………………………….… 12

4.5. Tournaments..…………………………………………………………………. 13

5. Poker…………………………………………………………………………………….. 14

5.1. Cash poker: general…………………………………………………………….. 14

5.2. Cash poker: a game of skill......…………………………………………….. 16

5.3. A skill analysis of tournament poker………………………………............. 16

References…………………………………………………………………………………. 22

87

Authors’ note

The original 2009 Dutch version of this report also contained a CD-ROM with 15 appendices to

facilitate its practical use at the time. Most of these appendices are listed in the references but are

unfortunately only available in Dutch. In any case they are not essential for understanding the main

line of argument of the report. In order not to deviate too much from the original version however

we have opted to maintain all references to the appendices in the current English version.

88

1. Introduction

The direct reason for this report was the request by Mr. W.M.C. van den Berg, examining magistrate

in charge of criminal matters in the district of Amsterdam, to answer the following question as part

of the BKB case (Blaas, Kurver and Blaas):

Why, in your view, should poker be treated as a game of skill, rather than as a game of chance?

First we provide a brief outline of our expertise. Ben van der Genugten is professor of Probability and

Statistics and Peter Borm is professor of Mathematics and Game Theory. We both work at the

Department of Econometrics and Operations Research of Tilburg University. Since 1990 we have

acted as expert witnesses in several court cases concerning the distinction between games of skill

and games of chance in the Netherlands and Austria. Several projects were performed jointly with

Marcel Das, professor of Econometrics and Data Collection and director of CentERdata, a research

institute affiliated with Tilburg University. Professor Das has reviewed a draft version of the current

report. We have also studied this topic intensively at a more theoretical level, through scientific

publications as well as through the supervision of PhD and Master students.

The current report builds on the conceptual framework of the general method of relative skill as

introduced in Van der Genugten & Borm (1994a), and explicitly draws on the simplifications and

refinements that this method has undergone over the past 15 years through both practical

experience and theoretical deliberations. In essence it has not changed, however; the method

provides a tool for the objective and consistent classification of games in terms of relative skill.

The classification method of relative skill has been developed and elaborated to serve Dutch law as

set forth in Article 1 of the Dutch Gaming Act: a game of chance is understood as

“an opportunity to compete for prizes or premiums, if the indication of the winners is subject to any form of chance, on which the participants generally cannot exercise any predominant influence.” In this description, the term opportunity stands for game in the widest sense of the word. Games in which money does not play a role are not relevant to the law on games of chance. We will therefore consistently assume that prizes and premiums are awarded in the form of money, and that the sums of money depend on the results of the game. The legal definition of a game of chance also alludes to chance elements. Apparently, to qualify as a game of chance, the game must contain some chance element that designates one possibility among a series of possibilities as the winning one, while this element generally cannot be influenced by the participants in any meaningful way. Such an element is usually referred to as a chance element or an uncertainty element, but often simply as “chance”. A recurrent theme in relevant verdicts is moreover the idea that a game is more skillful if the players can improve their

89

ability through study or through frequent practice. The game is then less likely to be qualified as a game of chance. We therefore designate a game as a game of skill if it is not a game of chance. Every game involving money is therefore either a game of chance or a game of skill.

Our general method regarding the level of relative skill of a game produces a number between 0 and

1, whereby the value of 0 corresponds with a pure game of chance (skill plays no role whatsoever)

and the value of 1 denotes a pure game of skill (in which chance plays no role whatsoever). This

specifically allows for a comparison and classification of games in terms of skill level, and hence to

determine the minimally required skill level (the skill threshold level), which will determine the

game’s classification as a game of skill.

In this report we choose to present our method and earlier conclusions in a fairly compact manner.

For a more extensive and detailed description and analysis, in several places we refer to the

appendixes on the enclosed CD-ROM containing integral versions of some of our earlier reports and

publications. In the text we therefore concentrate specifically on the main lines of argument leading

up to our earlier conclusions that a regular cash poker variant like Texas Hold‘em should be

considered a game of skill.

This report furthermore focuses on tournament poker, as this is the most frequently occurring form

of poker in the BKB case. It is not at all obvious to us that any tournament version of a game of skill

will automatically qualify as a game of skill as well. Accordingly, we have conducted a new and

specific analysis for a tournament model that is suitable both for the specific tournament involved in

the BKB case, and for the tournaments held regularly at Holland Casino. Given the time restrictions

under which this report was produced, it is a non-complex tournament model that should be viewed

as a reasonable, initial approach, which can be elaborated further through additional research. For

this reason we can only draw conclusions from this analysis with due caution. In paragraph 5 we

again keep to the main lines of argument. One of the appendixes on the CD-ROM contains a detailed

overview of the analyses and computer simulations of tournament poker, performed specifically for

this report.

Regarding poker variants, as far as we know only the fixed-limit cash poker variant has been assessed

under the Gaming Act. This is the first time that tournament poker is under assessment. One typical

feature of a tournament is that the prizes are awarded on the basis of an ultimate ranking of

participants, which is determined by the game results (generally in the form of tokens or points)

achieved during the rounds that make up the tournament. In this respect tournaments can well be

compared to the so-called management games, which have been legally assessed previously with a

view to the Gaming Act. In this study we were therefore able to benefit from our previous experience

with management games. It moreover turns out that tournament poker and management games are

comparable not only regarding the issue at hand, but also regarding the essence of our conclusions.

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The structure of the remainder of this report is as follows. Paragraph 2 presents our main conclusions. Paragraph 3 outlines existing Dutch case law regarding the Gaming Act, with particular attention for the state of affairs concerning poker. The situation concerning management games is also considered explicitly. Paragraph 4 describes the method of relative skill for single-player and multiple-player games, with reference to the scientific and practical validation of the method. The paragraph furthermore discusses on a general level how the relative skill of a cash game compares to the relative skill of this same game in a tournament context. Paragraph 5 sketches the considerations based on previous research to qualify common multiple-player cash variants of poker, such as Texas Hold’em, as a game of skill. Here we furthermore present a separate, quantitative analysis of tournament poker.

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2. Conclusions

Our conclusions issue from the method of relative skill. This method in principle enables a

classification of a large number of practical games in terms of their relative skill levels. However, it

does not answer the question exactly where to draw the skill threshold level. We have argued

previously that, with a view to consistency in case law, this threshold should be located between 0.1

and 0.3.

The classification of a game as one of chance or one of skill cannot automatically be made to apply to tournament versions of the same game. In analyzing the relative skill of tournaments, the number of participants in relation to the number of tournament rounds and the prize structure co-determine the outcome. Cash poker variants of Texas Hold’em should be classified as a game of skill, as their level of relative skill exceeds the 0.3 threshold. Our analysis of the Texas Hold’em tournament in the BKB case, explicitly taking into account the number of participants, the number of tournament rounds and the prize structure, does not give reason to classify this tournament version as a game of chance. The same conclusion applies to Texas Hold’em tournaments with around 50 participants, as organized for example by Holland Casino.

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3. Dutch case law

For a detailed overview of the drafting of the Gaming Act, and our commentary on the rulings and

verdicts regarding specific games until approximately the year 2000, please see Appendix 1 on the

CD-ROM.

3.1. The state of affairs regarding cash poker

Fixed-limit Texas Hold’em, along with other cash variants such as Five card draw poker, Omaha

Hold’em and Seven card stud poker, has been the subject of court cases. The first time that a

multiple-player game was made subject to the Gaming Act occurred in the district court of

Amsterdam, on 7 May 1996. The judge exonerated the organizers for a lack of evidence. As expert

witnesses in this trial, we presented our assessment system based on relative skill and recommended

that the poker variants be classified as games of skill. We emphatically pointed out the wide range of

strategic aspects involved in such multiple-player games, particularly where it concerns games that

involve incomplete information, as in this instance. This advice was followed by the court: “It is after

all conceivable that starting players quickly develop a certain measure of skill with which, combined

with other variables such as the strategy of other players, they can develop a personal strategy to

such an extent that it cannot be ruled out that they can generally exercise predominant influence

over the role of chance”. The court also adopted the view that not only chance elements such as the

cards one is dealt are essential to the poker variants, but also and especially the application of

randomized strategies by the players: “The in itself correct observation by the public prosecutor that

in the poker variants offered by the accused, a maximum of 41 to 47 (of the 52) cards may be

unknown at the end of the fifth round (just before the showdown), does not contradict the

foregoing. It does not force the conclusion that the participants cannot in general exercise

predominant influence over the chances of winning.”

This line of argument stands in stark contrast to that followed by the Court of Appeal and the

Supreme Court (in which we were no longer directly involved). The Supreme Court followed the

argumentation of the Court of Appeal. The Court of Appeal based its opinion on the argument that

the game rules directly determine that the impact of chance in (these variants of) the game of poker

is such that the players cannot have any influence. The game could only be considered a game of skill

if it can be demonstrated that this impact can be “overcome” through the use of some form of

probabilistic calculations. With this argument, the Court of Appeal ruled that the impact of chance

cannot be overcome. Here, the court followed the advice of an expert witness who stated that the

average player is not prepared to develop his skill as he only plays for the purpose of relaxation. This

is not a sound argument. What exactly does “overcome” mean? To overcome what exactly? The

argument presumes a type of player who simply performs a random lottery over all his possible

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actions at any possible decision moment. Yet no player will do so, also not in card games like

blackjack and bridge: given the structure of the game he will always pursue some form of strategy

with considerably better game results than by using a random lottery. In that sense alone he or she

more or less overcomes the impact of pure chance. To overcome should be replaced by “to do

significantly better than a beginner”. Here, the beginner is definitely another person than the above

described (non-existent) player using random lotteries all the time. The Court of Appeal relies on the

– to put it cautiously: -- non-verifiable judgment of an expert witness. We hold a different view. In

practice, poker players frequently play with a variety of other players. Our observation is that they

always strive to achieve a good result. This is quite typical, incidentally, for many multiple-player card

games, regardless of whether cash or game points are at stake. Finally, the Court’s general

deliberations are wholly directed at poker games, without considering the implications of any

comparative application of these deliberations to other card games such as bridge. In fact, it would

result in the erroneous conclusion that bridge is a game of chance rather than a game of skill.

The district court’s argumentation regarding the wealth of strategic aspects of poker variants is

annulled by subsequent imprecise deliberations by higher judges, who were possibly also

insufficiently cognizant of the difference in game characteristics between single-player games and

multiple-player games.

3.2. Management games

A recent ruling in proceedings concerns the so-called ‘management games’ of Competitie manager

and Grand Prix manager GPM (2 February 2005, no.105364, included as Appendix 2 on the CD-ROM).

These management games are operated via the internet and are the subject of a detailed analysis in

Van der Genugten, Borm & Dreef (2004), included as Appendix 3 on the CD-ROM, and in Van der

Genugten, Borm & Dreef (2005), Appendix 4 on the CD-ROM.

A typical feature of management games is that the final prize structure is determined by the final

ranking of all participants as determined by the game results (in points) achieved over the course of

the game rounds. In terms of structure, these games are thus comparable to tournaments. Since in

this case, the court basically adopted the arguments and conclusions of our reports in full, we shall

briefly discuss a few relevant details with regard to GPM here.

The goal of GPM is to compile a Formula 1 team in terms of car parts and staffing that will achieve

the best performance in a simulated season of Formula 1 competitions. It was possible to quantify

the role of chance in GPM on the basis of extensive data collection combined with statistical

techniques. In this way, the relative skill level was determined for various GPM variants that only

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differ with respect to the actual prize structure. Our first report concluded that, given a fairly

horizontal or gradually increasing prize structure (in which the prizes are not only awarded to a small

number of highly placed players in the final ranking), the relative skill level of GPM can be set at

around 0.3, so that the final verdict on the game is: game of skill. For less gradual prize structures (as

occurred in actual practice), the relative skill level came to a maximum of 0.1, so that the final verdict

on the game is: game of chance. Following the recommendations in our second report, the prize

structure of GPM was modified to create a more gradual structure, putting the relative skill level at

round 0.3 and assuring its legal classification as a game of skill.

How can these conclusions be explained in qualitative terms? The scores in GPM vary widely, but

there is a relatively large group of players who achieve scores that fall only slightly short of those

achieved by advanced players in top ranking positions. The small difference in scores amply remains

within the margin generated through chance. So if the prize structure is restricted to the players in

the top ranking segment, then chance plays an important role in the awarding of prizes. For if the

game is played repeatedly, then this relatively large group of players will often wind up as winners,

while an advanced player will not. This implies that the advanced players achieve a low game result

on average, with a relatively low learning effect as a result. In case of a more gradual or horizontal

prize structure, the results achieved by an advanced player will vary across repeated plays, but

distributed evenly at a high level. The learning effect will thus be greater in case of a gradual prize

structure.

3.3. Review and skill threshold level

If we combine the data regarding management games with previous legal rulings on explicit games of

chance and skill slot machines (cf. Van der Genugten, 1997a), then the designation of the skill

threshold level would reasonably be located between 0.1 and 0.3. Games with a level of relative skill

above 0.3 should in any case be classified as game of skill, and games with a relative skill level below

0.1 as games of chance. This method is in any case consistent with court rulings so far. The only

exception to this is the Supreme Court ruling following the Court of Appeal with regard to a number

of fixed-limit cash poker variants. It would have been preferable if they had followed the well-argued

judgment of the district court.

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4. Method of relative skill

The method of relative skill has been described extensively and meticulously in the general

publications by Hilbers, Hendrickx, Borm & Van der Genugten (2008), Dreef, Borm & Van der

Genugten (2004a, 2004b), Van der Genugten, Das & Borm (2001), Borm & Van der Genugten (2001),

Borm & Van der Genugten (1998), Van der Genugten (1997a), Van der Genugten & Borm

(1996a,1996b,1996c), and Van der Genugten & Borm (1994a).

Our method has been developed particularly with a view to studying so-called strategy games with

monetary rewards. Strategy games are about mental dexterity: the ability to make sensible decisions

systematically, which boils down to choosing a comprehensive game plan or strategy. A strategy thus

does not correspond to a general game attitude or approach, but it provides a detailed specification

of actions to be taken at any conceivable decision point in the game.

Strategy games can be classified according to a number of characteristic game features. A detailed

classification is provided in Van der Genugten and Borm (2005), as a contribution to a book written

for a legally versed audience. This contribution is included in full as Appendix 5 on the CD-ROM.

Game features are: the presence or absence of chance elements, the degree of complexity (e.g.

frequent or few decision moments), complete or incomplete information, equal or unequal

information among different players, and of course the number of players. As in more advanced card

games such as bridge, poker games typically involve multiple players, they contain chance elements,

they are highly complex, the information is incomplete, and there is an information disparity among

the players. Perhaps needless to say, but when referring to poker games we mean multiple-player

games such as Texas Hold’em, Seven card stud poker or Five card draw poker, and not single-player

games such as Caribbean stud poker or American poker.

In Paragraph 4.1 we describe in general terms the method of relative skill for strategy games with

chance elements. In Paragraph 4.2 we discuss the elaboration of this method for single-player

games. The main characteristic of a single-player game is that, although more players may be

engaged in playing the same game, any one player’s game result depends only on his or her own

chosen strategy, and not on any actions taken by other participants. Roulette, Golden Ten and

blackjack are typical examples. In multiple-player games such as Texas Hold’em and bridge, the result

achieved by each player typically depends partly on the decisions taken by other players. This makes

such games intrinsically much more complex. This interaction between players must be incorporated

in the method of relative skill for multiple-players games in an adequate and consistent manner.

Paragraph 4.3 explains how this is done. Paragraph 4.4 offers a practical and theoretical validation of

this method, drawing on relevant literature. Paragraph 4.5 offers a more general discussion of the

relationship between the relative skill of a strategy game and tournament versions of the same

game.

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4.1. General

Classification as a “game of chance” or a “game of skill’ depends on the relative skill of a game, as

determined by weighing two effects on the game result:

The learning effect (LE), due to the skill elements involved in the game, The random effect (RE), attributable to the chance elements involved in the game.

By “game result” we mean, in this context, the (probabilistic) expected gain: that is, the average gain

over an, in principle, infinite number of repeated game plays. This concretely concerns an amount of

money that is itself no longer dependent on chance. For games in which the stake amount is not an

intrinsic component of the strategy or game plan but more of a random choice in advance (as in

Roulette), we standardize the expected gains by dividing by the expected stake amount. Obviously,

the classification as game of chance or game of skill does not depend on the exact value for the game

result or on whether the game result is positive or negative for a certain strategy. We have opted for

the term learning effect since skill is achieved through study or experience. The random effect can

then be associated with any further improvement in terms of game result that could be achieved, if

the effects of the random factors were known in advance. The latter is of course a fictional situation.

The random effect offers a very elucidating means of measuring the variation in game results, solely

attributable to the chance elements involved in the game.

The relative skill of a game is expressed as a number, say S (“Skill”), which is large (maximum of 1) if the learning effect is dominant, and is small (minimum of 0) if the random effect dominates. A simple formula to express this, incorporating the terms of the underlying learning effect and random effect, is as follows:

S = LE / ( LE + RE). Thus, no learning effect (LE =0) yields S = 0 and no random effect (RE = 0) yields S = 1. Every game will always have a relative skill level of between 0 and 1, so that all games can be ranked in terms of their relative skill. 4.2. Single-player games The concepts of LE and RE have been made operational for single-player games by distinguishing three types of players: 1) the beginner, who plays in naïve manner, knows and understands the rules of the game but

lacks experience in actual play, with game result R(0), 2) the advanced player, who has mastered every aspect of the game, with game result R(m),

and

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3) the fictional (advanced) player, an advanced player in the theoretical situation that he knows the outcome of all chance elements during play (but cannot influence these), with game result R(f).

A specific player can learn the game and can become increasingly proficient at it, until he reaches the level of an advanced player. That is why the learning effect equals:

LE = R(m) – R(0). The extra result that the fictional player can attain over the result of the advanced players is solely attributable to the fictional knowledge of the outcome of the random factors. That is why the random effect equals:

RE = R(f) – R(m). For many casino games (such as Roulette and Golden Ten), the fictional player’s game result can simply be equated with the maximum result possible per game. For games with a small chance of a big win, only the fictional player will attain a high game result, so that the random effect RE is large and the relative skill, therefore, is small.

We wish to point out that the precise specification of the beginner’s strategy is a freely definable

parameter in our model. The skill level analysis can therefore be performed with any specification of

the beginner’s strategy. This means, specifically, that the fairly vague discussion about possible skill

elements in a game can be reduced to the more concrete question of what population of beginners

should function as reference point in determining the learning effect and the random effect.

When applying relative skill to specific games, it is necessary in each instance to adequately

characterize the three types of players. Experience shows that the application of the method of

relative skill to specific games requires a tailor-made analysis of the learning effect and random

effect, based on the particular features of the game. Also in the event of analytical restrictions, the

conceptual framework outlined above in any case provides a way of consistently comparing the

learning effect and random effect.

To get a better feeling for the method of relative skill for single-player games we refer to Appendixes

6 and 7 on the CD-ROM, in which the skill level analysis is illustrated with reference to the well-

known casino game blackjack and the less known but simpler game Spiel 21.

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4.3 Multiple-player games

In multiple-player games, the different player roles are generally asymmetrical. For poker, think for

instance of each player’s specific position at the table. We shall therefore start by examining a single

player in a specific player role, in the midst of several beginners as opponents.

We will portray this player as three different types: as a beginner, as an advanced player, and as a

fictional player, but each time playing against a group of beginners. This yields a specific game result

for each type. Both the advanced and the fictional type are presumed to be aware of the beginner’s

strategy for each player role. They will apply this knowledge to determine their own strategy.

We will then consider all the player roles for each of these three types of player, enabling us to

calculate the average game result across the player roles for each type of player: R(0) for the

beginner type, R(m) for the advanced type, and R(f) for the fictional type. The resulting figures serve

as input for the learning effect LE and the random effect RE, with which the relative skill S can be

determined.

It is a typical feature of card games that players do not know each other’s cards in many phases of

the game. In that sense they have incomplete and different information. This also applies to many

other multiple-player games. It can therefore be to one’s benefit to vary the decisions one takes in a

certain game situation. This can formally be portrayed as a randomized choice between certain

possible actions with individually chosen probabilities. This internal decision-making mechanism, not

perceptible to other players, we describe as randomized. A strategy containing such randomized

decision-making mechanisms is accordingly referred to as a randomized strategy. For a closer

examination of the role and meaning of randomization, we refer to the discussion on bluffing and

sandbagging in the specific context of multiple-player poker variants in Paragraph 5. Players that

randomize in this internal way add their own chance elements to the game. In our analysis we

therefore assume that the fictional player also knows the outcome of these internal lottery

mechanisms.

To determine the skill level, the point of departure is again the beginner’s strategy. This strategy

must be established for every player role and may in principle be randomized. This specification can

again be considered as a freely definable parameter in our model. The analytical restrictions for

concrete multiple-player games are much larger than for single-player games. In any case, the

methodical approach again offers a useful conceptual framework for the consistent weighing of the

learning effect and random effect.

For a more detailed conceptual discussion on relative skill in multiple-player games we refer to

Hilbers, Hendrickx, Borm & Van der Genugten (2008), included on the CD-ROM as Appendix 8.

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4.4. Theoretical and practical validation

First developed in Van der Genugten and Borm (1994a) and subsequently refined over the next 15 years, the method has amply proved its scientific value and practical applicability as an objective and consistent means of classifying games based on their relative skill.

Scientific output

Relative skill is the subject of the dissertations “Skill and strategy in games” (Dreef, 2005) and

“Golden ten and related trajectory games” (De Vos, 1997), which were both successfully defended at

Tilburg University. The method is furthermore addressed in two recent graduation theses, by Maaten

(2009) and by Hilbers (2007). Scientific reports and publications relating to the method of relative

skill are: Van der Genugten (2008, 2003, 1997c, 1993), Hilbers, Hendrickx, Borm & Van der Genugten

(2008), Dreef & Borm (2006), Dreef, Borm & Van der Genugten (2004a, 2004b, 2003), Borm & Van

der Genugten (2001, 1998), Van der Genugten, Das & Borm (2001) and Van der Genugten & Borm

(1996c).

Practical applications

The method of relative skill has been applied to a large number of actual games, on the request of

the public prosecutor, defense lawyers and/or the examining magistrate, as part of judicial

proceedings: Blackjack: Van der Genugten & Das (1999b), Fruitcard, Carribean Studpoker en

American poker 2: Van der Genugten & Das (1999a, 1999c, 1998), ROTA-roulette: Van der Genugten,

De Vos & Das (1999), Dromus-24: Van der Genugten & De Vos (1998), Jokeren, Jahtzee en

Eurobsgame: Van der Genugten (1997b), Concard Aces: Van der Genugten (1996), Piramidesystemen:

Van der Genugten & Borm (1994b), Spot the balll: Van der Genugten & Brekelmans (2006), Random

Flashback: Van der Genugten & Das (2005), Managementspelen: Van der Genugten, Borm & Dreef

(2005, 2004), and, finally, Poker: Borm & Van der Genugten (2005) and Van der Genugten, Borm &

Grossmann (1997).

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4.5. Tournaments

So far, tournaments have not formally been the subject of Dutch cases of law. There has been a case

of Swedish case of law in which a tournament version of Texas Hold'em was classified as a game of

skill. We quote the key formulation: “…in this version of poker it is the players’ actual aptitude and

analytical capabilities that are the winning point and not the actual luck factor”. For further

information we refer to Appendix 9 on the CD-ROM.

Without performing an explicit skill level analysis, in this paragraph we wish to already discuss a few

general aspects that play a role in the classification of tournaments. Virtually every game can be

organized as a tournament, with the actual game functioning as separate game rounds. There are no

grounds, however, for equating “game of chance (or of skill) game round” with “game of chance (or

of skill) tournament”. The classification of a tournament very much depends on its design.

Take, as an example, a game in which each player’s goal is simply to win. The winning probability of the advanced player is greater than the winning probability of a beginner. The qualification as a game of chance or of skill also depends on the winning probability of the fictional player. These probabilities determine the relative skill S of a single play of the game. Now suppose this game were to be played in a tournament form, in which all players consistently participate in each game round, and all the game rounds are independent of each other. The tournament winner is the player who wins the largest number of game rounds, and only this person is awarded a prize. In this tournament form, the (tournament) winning probability of the advanced player increases in tandem with the number of game rounds. The game may continue to be a game of chance if the number of game rounds is limited. However, if the tournament consists of a large number of game rounds, then the tournament prize ultimately will be won by the advanced player, and the tournament will be (almost purely) a game of skill. This always applies, regardless of how small the difference between the winning probabilities per tournament round of the advanced player and the beginner. Let us now adapt the rules of the tournament such that, after a fixed number of rounds, only those players that have won a certain number of the game rounds may progress to the next round. This introduces the possibility that an advanced player will be eliminated after a certain number of game rounds, and thus does not win a prize. This person’s probability of winning the tournament is therefore limited, even if the tournament consists of a very large number of game rounds. This design of the tournament means that the level of relative skill remains limited with an increasing number of game rounds. Finally, we return to the original tournament design (no eliminations), but we introduce a prize structure that corresponds strictly with the number of game rounds won. Compared to ‘only the tournament winner gets the prize’, for the same number of tournament rounds the relative skill of this new type of tournament will be higher. After all, the advanced player will receive a prize more often than the beginner, even if he does not end up as the tournament winner. For a game in tournament form, the number of game rounds and the prize structure thus play an important role. In fact we already encountered this phenomenon in Paragraph 3.2, with respect to management games.

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In practice, game rounds are usually not independent of each other. Maximizing the result per game

and achieving the highest possible final ranking in a tournament version of the same game are

different goals, then. This implies, in particular, that good or advanced strategic behavior in a game

and in a tournament version of the same game can be an intrinsically different matter, and hence

that a game and its tournament version can differ in terms of relative skill. A straightforward example

of this is provided in Maaten (2009), showing that the relative skill levels of a two-player cash poker

game and a tournament version (aimed at winning more tokens than the opposite player and a fixed

sum for the one who wins most tokens) can differ considerably.

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5. Poker

5.1. Cash poker: general

Among the more common multiple-player cash poker variants such as Seven card stud, Five card

draw poker and Texas Hold’em, the latter game is most popular in the Netherlands. Each game has

different betting structures: fixed-limit, pot-limit or no-limit.

In earlier, general publications such as Van der Genugten (2008, Appendix 10 on the CD-ROM) and

Van der Genugten & Borm (2005, Appendix 5 on the CD-ROM), it was shown how skill in poker

variants is closely tied to practical skills regarding the calculation of probabilities, which play a role in

evaluating the quality of one’s own cards in combination with information from the flop, or in

evaluating the quality of other players’ cards. This calculation of probabilities is described in the

many books written about cash poker, discussing the various rules of thumb to determine one’s

expected gains. The quality of these calculations can be appraised through computer simulations.

The Pokerstove program, included as Appendix 11 on the CD-ROM, is a good example.

Besides skills in terms of evaluating one’s options through calculations of probabilities, more

psychological techniques such as bluffing (bidding on a bad hand, instead of folding) and sandbagging

(not bidding on a good hand, but going along or checking) form an essential part of good game

strategy. Both techniques aim to make the best possible use of the fact that all of the players,

without exception, are in a situation of incomplete information (nobody knows another player’s

cards); this is something that inexperienced poker (but also bridge) players tend to realize

insufficiently. The best way to apply these techniques can, in theory, be calculated mathematically.

Naturally, players should not always bluff or use a sandbagging type of strategy, as this would

neutralize the uncertainty factor. This leads to randomization, in quite a natural manner: a good

bluffing or sandbagging strategy uses an internal randomization mechanism. As noted above, the

odds to apply here (how often should I bluff with this type of hand?) can be exactly calculated in

principle, but players can also develop a good feel for this simply through frequent play. In that sense

they learn to play the game, and skill is acquired. There is an extensive literature on this subject.

There is even special software to help develop such strategies through self-study; see for instance

the computer program “Turbo Texas Hold‘em for Windows”, as described in Wilson (2005).

To characterize a beginner’s strategy, it seems reasonable to assume that a beginner does not apply

the techniques described in various poker handbooks. After all, studying and mastering these

techniques requires a considerable effort, which the beginner has yet to make. An advanced player

can of course be assumed to have made this effort. In accordance with this line of reasoning, the

typical ingredients of a beginner’s strategy would be: to stay in the game for too long with relatively

poor cards (no or wrong estimation of probabilities), and not (successfully) applying the techniques

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of bluffing and sandbagging (too little game experience). The general import of these ingredients has

been confirmed to some extent by the results of the Texas Hold’em experiment, as described in

Maaten, Borm, Van der Genugten & Hendrickx (2008), included as Appendix 12 on the CD-ROM. This

experiment clearly demonstrated that many beginners in any case wish to stay in the game until the

flop, that they do not use sandbagging techniques, and that they attempt to bluff sporadically,

without careful timing. In our skill level analysis concerning cash poker, as reported in various

publications, these ingredients will always, in one way or another, form the foundation for the

choices made in a beginner’s strategy, which, as argued before, serves as reference point for the

determination of relative skill.

We conclude this paragraph with a general comment on the possible difference in relative skill

between fixed-limit, pot-limit and no-limit variants of multiple-player poker. The results by Hilbers

(2007) and Hilbers, Hendrickx, Borm & Van der Genugten (2009) concerning the variation in the

height of the possible fixed-limit bidding from low to high, indicate that the difference in relative skill

between the three variants is not significant. Restricting the skill level analysis to fixed-limit variants

thus appears justified.

5.2. Cash poker: a game of skill

The report entitled “Poker: ein Geschicklichkeitsspiel!” (Borm & Van der Genugten, 2005, Appendix

13 on the CD-ROM) extensively argues that, with a view to consistent jurisprudence, common

multiple-player cash poker variants such as Texas Hold’em should be classified as games of skill, on

the basis of a relative skill level analysis. This conclusion was based on both an exact quantitative

analysis of relatively simple stylized poker variants that did however incorporate key skill aspects of

real poker variants, and on a more qualitative analysis based on a comparison of so-called skill

indicators.

More recent studies into complex poker variants that more closely approximate the real poker game,

by means of straight poker in Hilbers (2007) and Hilbers, Hendrickx, Borm & Van der Genugten

(2009), once again confirm this conclusion. Depending on the exact specifications of the game

(number of players, bet heights, number of bidding rounds, the rake, choice of beginner’s strategy),

the relative skill level varies consistently between 0.3 and 0.5. A further confirmation that the skill

level is at this level is provided by the simulation results regarding Texas Hold’em, with two players

and two phases (“pre-flop” directly followed by “river”); see Appendix 14 on the CD-ROM.

In realistic poker variants, there is at least one additional source that demonstrates skill, compared to

straight poker or the just-indicated two-phase poker. The point is that the ultimate composition of a

player’s hand of five cards is built up in a larger number of phases, possibly with open own cards or

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shared (community) cards. Generally speaking, in a complex game with various phases an advanced

player will have more scope to obtain information – information which is freely available to the

fictional player. In a global sense we could thus say that, in a more complex game, the difference in

information between an advanced and a fictional player plays a comparatively smaller role, which

means that the relative skill level is greater.

5.3. A skill analysis of tournament poker

The study into the relative skill level of multiple-player cash poker reveals that, depending on the precise specifications, this game has a skill level ranging between 0.3 and 0.5. Tournament poker differs from cash poker in the following essential manner. Players purchase tokens, which are the only stake throughout the tournament, and the ultimate number of tokens won determines players’ final ranking. Payouts are subsequently awarded on the basis of this ranking only. The relative skill of tournament poker therefore comes to depend in part on the prize structure, and as argued in paragraph 4, this skill level can differ from that of the underlying cash poker game. To obtain insight into the relative skill level of tournament poker, we assume a stylized “base form” of a single game round. If such a single game round is only played for cash, then this game corresponds with cash poker. On account of the limited time available to us, the chosen base form of a single play round is a somewhat crude model, but this could be refined in further research. Tournament poker consists of a series of such stylized game rounds, with tokens at stake. The tokens determine the final ranking, which then translates into cash payouts in accordance with the prize structure. Given our choice for this base form, we studied a number of base tournament variants which explicitly incorporate the characteristic aspects that play an effective role in practice. The step taken in our model from a game round to a tournament is thus an accurate reflection of actual practice. The model particularly takes into account the number of players, the anticipated duration of the tournament, the number of game rounds, the changes of table, the variation in the number of players per table due to possible quitters, the initial amount of tokens and the stake increases across the game rounds. The only aspect not factored in is the dependency between game rounds, generated by a game strategy that stretches across the entire tournament. To incorporate this dependency would make the analysis hugely complex and time-consuming. The so-called BKB base tournament models the BKB poker tournament with 45 participants. Unfortunately, the information available to us did not indicate the exact prize structure. However, it appears from several witness accounts that this structure does not differ significantly from the prize structure used in comparable poker tournaments in Holland Casino, and thus we have used this structure in our analysis. We additionally performed a specific analysis with regard to the so-called HC base tournaments that model the poker tournaments as organized by Holland Casino, based on the Master Classics of Poker. In our analysis we specifically vary the number of players within a realistic duration of play. A more detailed description of the simplified form of a single game round and the corresponding tournaments is offered in the second part of Appendix 15 (poker tournaments). We shall restrict our discussion to the main points here. An important quantity in the analysis concerns the winning percentage (Win%) of the bet that an advanced player will achieve during a base game round with a single beginner as opponent. In our simplified model, this winning percentage should correspond with the winning percentage that

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an advanced player would achieve at a real cash poker table with a single beginner as opponent. This winning percentage may possibly be derived as well from the real winning percentage achieved by an advanced player at a cash poker table with multiple beginners. Ideally, an accurate estimate of the winning percentage Win% should be derived from empirical material. However, given the nature of this quantity – an advanced player among beginners – such material is not immediately available. The only usable data available derives from the Poker Experiment as documented in Maaten, Borm, Van der Genugten & Hendrickx (2008, Appendix 12). However, given the statistical limitations of the experiment on account of the small number of observations, the analysis contained in the first part of Appendix 15 (bids and gains) only indicates a lower limit for the winning percentage Win%. This percentage will in any case be higher than 5%, and will probably be considerably higher, but we wish to be cautious about drawing any clear-cut conclusions on the basis of this limited experiment. If we take the following rule of thumb (see Meinert (2007), page 270, a standard reference book on poker): “A skilled, solid player places approximately two big bets per hour. So if you play 3 / 6 Euro fixed-limit poker, the hourly wage comes to … around 12 Euros” -- and through some mathematical operations translate this to correspond with our base model, then this means a percentage of 3.5%. It should be noted that the quoted rule of thumb assumes a regular, real-life poker table; so instead of having one “skilled, solid player” facing only beginners, there will be other experienced or even advanced players at the table. The 3.5% derived from the rule of thumb thus signifies a lower limit for the winning percentage Win%. This percentage will in fact be considerably higher than 3.5%, and the percentage of 5% determined through the poker experiment would appear to be a cautious lower limit, based on the quoted rule of thumb. In our analysis we therefore only consider winning percentages starting from 5%. Further, the relative skill level (S-cash) of the corresponding cash poker should be located between 0.3 and 0.5, to remain in conformity with our previous findings for cash poker variants. For all base forms that meet the conditions above, we subsequently calculate the relative skill level of the associated base tournament in four different ways: S-1, S-2, S-3 and S-4. The difference between them is due to the different choices with regard to the prize structure. We have opted to do so in order to make visible the effect of the prize structure on the relative skill. We will now explain the differences between the four calculation methods. S-1: the level of relative skill that is based on a prize structure in which only the final tournament winner receives a payout. S-2: the level of relative skill that is based on a gradual prize structure in which all players receive a payout proportional to their final ranking. With 45 participants, this leads to the structure [45 44 43 … 3 2 1 0 … 0]. S-3: the level of relative skill that is based on a prize structure in which all players at the final table receive a payout proportional to their final ranking while the other players do not receive any payout. With 9 participants at the final table, this leads to the structure [9 8 7 6 5 4 3 2 1 0 … 0]. S-4: the level of relative skill that is based on the prize structure as used by HC and BKB. The 6 best players at the final table in the final ranking receive payouts in a less gradual way (than with respect to S-3) to their final ranking while the other players do not receive any payout. This leads to the structure [37 23 15 11 8 6 0 … 0]. For the classification of the BKB and HC tournaments, the relative skill S-4 is essential given the fact that the associated prize structure most resembles the prize structure that is actually used in these tournaments. The relative skill levels S-1, S-2 and S-3 not only serve for comparison but also to verify the qualitative arguments with respect to the form of the prize structures, as indicated in paragraph 3.2 with respect to management games. The relative skill level S-1 corresponds with the most extreme prize structure imaginable: only the tournament winner wins a prize. On the opposite side of the spectrum, S-2 corresponds with a very gradual prize structure that is fully

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proportionate to a player’s final position in the final ranking. We shall see that, also for base tournaments, the relative skill level S-1 (corresponding with an extreme prize structure) turns out significantly lower that skill level S-2 with a gradual prize structure. Relative skill levels S-3 and S-4 correspond with prize structures that might be considered intermediate forms, with the prize structure that corresponds with S-3 having a more gradual structure than the structure for S-4. To give an idea of our simulation results for the BKB base tournament, a representative amount of data is summarized below in Table 1. The full version is given in the second part of Appendix 15 (poker tournaments).

Table 1. Relative skill levels of poker tournaments, depending on the winning percentage (Win%) and the prize structure.

Win% S-cash S-1 S-2 S-3 S-4

5 0.50 0.35 0.51 0.44 0.40

5 0.45 0.30 0.44 0.38 0.35

5 0.42 0.26 0.41 0.35 0.31

5 0.38 0.23 0.39 0.32 0.28

5 0.36 0.19 0.34 0.27 0.24

5 0.33 0.17 0.32 0.25 0.22

5 0.31 0.15 0.31 0.24 0.20

10 0.50 0.28 0.47 0.39 0.34

10 0.45 0.24 0.43 0.35 0.30

10 0.43 0.22 0.41 0.32 0.27

10 0.37 0.17 0.36 0.27 0.22

10 0.35 0.15 0.34 0.25 0.20

10 0.32 0.14 0.32 0.23 0.19

10 0.30 0.13 0.31 0.23 0.18

13 0.50 0.28 0.48 0.39 0.34

13 0.46 0.25 0.45 0.36 0.31

13 0.43 0.23 0.43 0.34 0.28

13 0.38 0.20 0.40 0.30 0.25

13 0.36 0.19 0.39 0.29 0.24

13 0.32 0.18 0.37 0.28 0.23

13 0.30 0.17 0.36 0.27 0.22

15 0.50 0.30 0.53 0.43 0.37

15 0.45 0.27 0.50 0.40 0.33

15 0.43 0.26 0.48 0.38 0.32

15 0.39 0.24 0.46 0.36 0.30

15 0.35 0.23 0.44 0.34 0.29

15 0.33 0.23 0.44 0.34 0.28

15 0.31 0.22 0.43 0.34 0.28

17 0.50 0.33 0.53 0.45 0.39

17 0.46 0.31 0.51 0.43 0.37

17 0.44 0.30 0.50 0.41 0.36

17 0.39 0.29 0.49 0.40 0.35

17 0.35 0.28 0.48 0.39 0.34

17 0.33 0.28 0.47 0.39 0.34

17 0.30 0.28 0.47 0.39 0.33

20 0.50 0.40 0.52 0.46 0.43

20 0.45 0.38 0.50 0.44 0.41

20 0.43 0.37 0.48 0.42 0.39

20 0.40 0.36 0.47 0.41 0.38

20 0.35 0.35 0.45 0.40 0.37

20 0.33 0.35 0.45 0.40 0.37

20 0.30 0.35 0.45 0.39 0.37

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For illustration purposes, we shall examine the following row in this table: Win% S-cash S-1 S-2 S-3 S-4

15 0.39 0.24 0.46 0.36 0.30

This row indicates that, when the fixed winning percentage equals 15% and the fixed relative skill level S-cash = 0.39 in the base form of the cash poker game, the relative skill level S-4 of the corresponding BKB base tournament can be determined as 0.30. A more gradual prize structure for the final table results in a higher relative skill level of S-3 = 0.36, and a fully gradual prize structure results in an even higher skill level of S-2 = 0.46. However, with a payout for only the winner, the relative skill level would result in a lower figure: S-1 = 0.24. To analyze the influence of the winning percentages, in Table 1 we consider 6 different winning percentages: 5%, 10%, 13%, 15%, 17% and 20%. For each of these percentages we consider seven values for the relative skill level S-cash for the base form of cash poker, which adequately cover the potential range from 0.3 and 0.5. From the last column in Table 1, listing the relative skill level S-4, we may conclude that our analysis does not support the classification of the BKB poker tournament as a game of chance. The relative skill S-4 is mostly above 0.2 for all winning percentages, and mostly above 0.3 for a winning percentage of 15% or higher. Remember also that, in choosing the winning percentage, we assumed a real minimum level of 5%. More theoretical considerations such as the analysis for two-phase Texas Hold’em in Appendix 14 suggest that Win% may well exceed 20%, so that it seems entirely justified to classify this a game of skill. However, exercising due caution with respect to our rough modeling, we prefer to restrict our conclusion to saying that there is “no reason for classification as a game of chance.” If we compare the level of relative skill associated with the different prize structures, we see that the BKB base tournament, in which only the tournament winner is awarded a payout with a winning percentage of up to approximately 15%, mostly results in skill levels (see column beneath S-1) of above 0.1 and below 0.3, so that the skill classification remains unclear. However, with a winning percentage of 20% the skill levels would amply exceed 0.3 for this prize structure as well. If the fully gradual prize structure that corresponds with the relative skill levels in the column beneath S-2 were to be applied, then the classification as a game of skill is clear. Mainly on account of the low number of participants of 45, this would also apply in case of the more gradual prize structure at the final table, corresponding with the relative skill levels as listed in the column beneath S-3. The above implies that, for the BKB poker tournament to be classified even more clearly as a game of skill, it would be advisable to adopt a somewhat more gradual prize structure than currently applied. An interesting ancillary conclusion from the above results is that the statement “tournament poker involves more skill than cash poker”, or “cash poker involves more skill than tournament poker”, both prove to be incorrect. With a winning percentage of 17% and a skill level for cash poker of 0.39, the relative skill level S-4 of the corresponding tournament equals 0.35, whereas, with a relative skill level for cash poker of 0.30, the relative skill level of the corresponding tournament equals 0.33. The analysis with respect to the HC base tournaments yields a picture comparable to that for BKB base tournaments, also regarding the important effect of the prize structure on the level of relative skill. On the basis of our findings, there is no reason to classify an HC poker tournament with 50 participants as a game of chance. Our analysis does show very clearly, however, that the number of participants in a tournament in relation to the duration of the tournament plays a crucial role in the skill classification. Already with 100 participants, the prize structures that only award prizes to players on the final table result in relative skill levels (S-3 and S-4) that predominantly fall below 0.3. With 250 or 500 participants, this effect is only amplified. Where such variants are concerned, classification as “game of chance” would seem more appropriate.

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Tilburg University.

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Dutch), Tilburg University.

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behendigheidsspel? (& repliek). Reports (in Dutch), Tilburg University.

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diagnostische criteria. Report (in Dutch), Tilburg University.

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Behendigheidsniveau van Texas Hold’Em

Een experiment uitgevoerd in samenwerking met VARA-Nieuwslicht

Concept-verslag

Rogier Maaten

Peter Borm, Ben van der Genugten, Ruud Hendrickx

Universiteit van Tilburg, 27 maart 2008

Inhoudsopgave

1. Inleiding 2

2. Fixed Limit Texas Hold’Em 2

2.1 Notatie 2

2.2 Pokerhanden 2

2.3 Spelverloop 3

2.4 Behendigheidsniveau 5

3. Opzet en uitvoering experiment 5

3.1 Doel experiment 5

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3.2 Globale opzet experiment 6

3.3 Details van de opzet van het experiment 6

3.4 De voorbereidende fase 6

3.5 De aanloopfase 7

3.6 De speelfase 7

3.7 Tafelindeling 7

3.8 Uitvoering 8

4. Analyse experiment 9

4.1 Spelverlopen 9

4.2 Resultatenlijst 9

4.3 Variantieanalyse resultaten 10

4.4 Behendigheidsniveau 10

4.5 Beginnersstrategie 12

4.6 De ervaren speler 17

5. Conclusies 17

6. Bijlage: Spelverlopen 18

7. Bijlage: Resultaten per positie 68

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1. Inleiding

Volgens de Hoge Raad is poker een kansspel. Hierdoor mag poker officieel uitsluitend gespeeld

worden in vestigingen van Holland Casino; commerciële exploitatie op andere locaties is niet

toegestaan. Is deze classificatie terecht, of is poker eigenlijk een behendigheidsspel?

In een experiment, gehouden in samenwerking met de VARA-Nieuwslicht, is getracht om het

behendigheidsniveau van poker aan het licht te brengen. De details van het experiment, en de

resultaten ervan zijn in dit rapport te vinden.

De opbouw van dit rapport is als volgt. Allereerst worden de spelregels en het spelverloop van Texas

Hold’Em beschreven, gevolgd door een beschrijving van het pokerexperiment. In het hoofdstuk

“Analyse experiment” wordt een schatting van het behendigsheidsniveau van poker gegeven aan de

hand van de resultaten van het experiment. Ook is er een karakterisering van de strategie van de

beginners te vinden. Afgesloten wordt met enkele conclusies betreffende de nauwkeurigheid van de

resultaten van dit experiment en het behendigheidsniveau van Texas Hold’Em.

2. Fixed Limit Texas Hold-Em

2.1 Notatie

De verschillende kaarten in het spel worden als volgt aangeduid: A,K,Q,J,T,9,8,7,6,5,4,3,2. Hierbij

staat A voor ace (aas), K voor king (koning), Q voor queen (vrouw), J voor jack (boer) en T voor ten

(tien). De getallen in deze reeks spreken voor zich.

Een pokerspel bevat vier kleuren: schoppen (s van spades), harten (h van hearts), ruiten (d van

diamonds) en klaveren (c van clubs).

Een speler die AhJs ontvangt heeft dus de aas van harten en de boer van schoppen.

Wanneer een hand twee kaarten van dezelfde kleur bevat, dan noemen we deze hand “suited”.

Dit wordt genoteerd als A9s, 74s, etc. Handen die niet suited zijn, noteren we als T9o, 52o (o staat

voor “off-suit”).

2.2 Pokerhanden

Texas Hold-Em lijkt de meest populaire vorm van poker. Het spel wordt gespeeld met minimaal twee

en maximaal tien spelers. Iedere speler krijgt twee kaarten, die niet aan de andere spelers getoond

worden. Deze kaarten worden gecombineerd met vijf kaarten, die open op tafel komen te liggen

gedurende het spel (genaamd “community cards”). Van de twee kaarten in de hand en de vijf

community cards gebruikt iedere speler in totaal vijf kaarten om daarmee een zo sterk mogelijke

pokerhand te maken.

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De pokerhanden van hoog naar laag zijn:

Royal Flush: AHVBT in een kleur. In poker zijn alle kleuren van gelijke hoogte. In andere varianten van

poker (bijvoorbeeld 7-Card Stud) is het mogelijk dat twee spelers een royal flush halen in een spel.

Straight Flush: Vijf opvolgende kaarten in een kleur, bijvoorbeeld 98765 van harten. Een aas mag

gebruikt worden als hoogste kaart in het spel, maar ook als laagste kaart in het spel. 5432A van

ruiten is dus ook een straight flush, 32AKQ van klaveren is geen straight flush. Tussen straight flushes

onderling is de hoogste kaart de beslissende factor. 98765 wint dus van 5432A.

Four of a kind: Vier kaarten van dezelfde rang, zoals vier vrouwen. Ook wel bekend onder de naam

“quads”. De rang beslist in het geval dat meerdere spelers four of a kind hebben, dus KKKK3 wint van

5555A. Als meerdere spelers een vierkaart van dezelfde rang hebben, dan is de hoogste bijkaart

cruciaal. Dit komt voor wanneer van de vijf community cards er vier dezelfde rang hebben. Als TTTT6

op tafel ligt, speler A heeft QJ in de hand en speler B heeft A7, dan heeft speler B de beste hand

(TTTTA).

Full House: Combinatie van drie kaarten van dezelfde rang plus twee kaarten van een andere rang,

bijvoorbeeld JJJ44. De rang van de driekaart is beslissend indien er meerdere spelers een full house

hebben gemaakt. 999QQ wint van 666KK.

Flush: Vijf kaarten in een kleur, zoals K9632 van schoppen. De hoogste kaart is bepalend bij flush

tegen flush, daarna de één na hoogste, etc.

Straight: Vijf opvolgende kaarten, zoals QJT98. Kleur is niet van belang. De hoogste kaart zorgt weer

voor het verschil als een andere speler ook een straight heeft gemaakt.

Three of a Kind: Drie kaarten van dezelfde rang plus twee andere kaarten, bijvoorbeeld 888KT. De

rang bepaalt weer bij meerdere three of a kinds, bij gelijke rang beslist de hoogste bijkaart. Andere

benamingen voor three of a kind zijn “trips” of een “set”.

Two Pair: Een pair bestaat uit twee kaarten van dezelfde rang. Two pair is een combinatie van twee

verschillende pairs, aangevuld met met de hoogst overgebleven kaart. Het hoogste pair is cruciaal in

geval van two pair tegen two pair. Daarna beslist het tweede paar, en daarna de bijkaart. AA882

verslaat KK55Q, en 9944K verslaat 99226.

Pair: Dit is een combinatie van een pair en drie overige kaarten. Zoals gebruikelijk is 77K83 beter dan

55AKQ, en wint KKQ65 van KKJ95.

High Card: Hieronder vallen alle overige handen. De hoogste kaart is cruciaal als het gaat om high

card versus high card.

2.3 Spelverloop

De spelers zijn beurtelings deler (of “dealer”). Met zes spelers aan tafel is iedere speler dus om de zes

spellen dealer. De speler direct links van de dealer plaatst een verplichte inzet van 1 fiche aan het

begin van het spel. Deze inzet wordt de small blind genoemd (kortweg SB). De volgende speler, twee

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plaatsen van de dealer, plaatst de big blind (BB) van 2 fiches. Vervolgens worden de kaarten gedeeld,

en krijgt iedere speler zijn twee kaarten in de hand. Aan de hand hiervan, en aan de hand van de

acties van de andere spelers beslissen zij welke actie ze zullen ondernemen. Ze hebben de volgende

opties:

Fold: Terugtrekken uit het lopende spel. De kaarten worden teruggegeven aan de dealer, en de

speler speelt verder geen rol meer in het spel.

Call: De call-optie gebruik je indien je door wilt spelen in de hand. Je investeert 2 fiches in dit spel

(evenveel als de big blind). Mocht er al door een andere speler zijn verhoogd, dan kost callen meer

fiches (evenveel als hoeveel de speler die heeft verhoogd heeft ingezet).

Bet/Raise: De huidige inzet verhogen. De eerste verhoging wordt altijd aangeduid met “bet”, alle

volgende verhogingen zijn “raises”. Aangezien we Fixed Limit Texas Hold’Em spelen, kunnen spelers

uitsluitend verhogen met 2 fiches. Het maximaal aantal verhogingen in deze spelronde is vier, spelers

kunnen in deze fase dus niet meer dan 8 fiches investeren.

Check: De check-optie is in deze fase alleen relevant voor de big blind in het geval dat niemand

verhoogt. Hij heeft al zijn verplichte 2 fiches geinvesteerd, en heeft nu de optie om zonder extra inzet

door te spelen.

Een voorbeeld van mogelijke acties staat weergegeven in Tabel 1. De vetgedrukte speler, A6, is de

dealer. Speler A1 heeft de small blind geplaatst en speler A2 de big blind. Spelers A4 en A5 hebben

zich teruggetrokken uit de hand, de overige spelers hebben ieder 2 fiches geïnvesteerd, en spelen nu

verder. Niemand heeft verhoogd.

A1 A2 A3 A4 A5 A6

SB BB Call Fold Fold Call

Call Check

Tabel 1: Acties op de preflop

Als alle acties zijn afgerond, komen de eerste drie open kaarten (van de vijf community cards) op

tafel te liggen. Samen heten deze drie kaarten de “flop”. Analoog aan de vorige betronde hebben de

overgebleven spelers nu weer de mogelijkheid om in te zetten. De speler op de positie van de small

blind is weer als eerste aan de beurt. Hij heeft de volgende opties: check en bet. Tot er een speler

een bet plaatst, hebben de spelers dezelfde opties: check en bet. Als een speler een bet heeft

geplaatst, hebben de volgende spelers deze opties: fold, call en raise. Weer geldt dat vier bets het

maximale aantal is, nadat iemand een bet heeft geplaatst mag er dus nog maximaal drie keer worden

geraised. Een bet bedraagt nog altijd 2 fiches.

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Een mogelijk biedverloop op de flop is weergegeven in Tabel 2 (waarbij spelers A4 en A5 zich al voor

de flop, oftewel preflop, hebben teruggetrokken). Speler A2 houdt het nu ook voor gezien, de andere

drie actieve spelers hebben allen 4 fiches geinvesteerd in deze fase.

A1 A2 A3 A4 A5 A6

Check Check Bet Call

Raise Fold Call Call

Tabel 2: Acties op de flop

Nu komt de vierde open kaart op tafel, dit wordt de “turn” genoemd. Deze fase is identiek aan de

“flop”, behalve dat iedere bet nu 4 fiches kost in plaats van 2. Het voorbeeldspel gaat verder als

aangegeven in Tabel 3.

A1 A2 A3 A4 A5 A6

Check Bet Call

Fold

Tabel 3: Acties op de turn

Speler A1, die op de flop nog verhoogd heeft, geeft hier de hand op. Wellicht was de turnkaart voor

hem ongunstig, of hij was aan het bluffen en verloor de hoop dat zijn tegenspelers de hand op

zouden gaan geven. Spelers A3 en A6 gaan verder naar de “river”.

De “river” is de laatste fase in het spel. De laatste van de vijf community cards wordt nu getoond. De

resterende spelers zien nu welke hand ze gemaakt hebben. De betronde is hetzelfde als op de turn,

een bet kost 4 fiches. Speler A3 heeft nogmaals verhoogd. Speler A5 erkent zijn verlies en is niet

bereid om nog 4 extra fiches in deze hand te investeren, zoals weergegeven in Tabel 4.

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A1 A2 A3 A4 A5 A6

Bet Fold

Tabel 4: Acties op de river

Aangezien speler A3 de enige overgebleven speler is, ontvangt hij nu de inzetten van alle spelers in

dit spel. Hij wint dus de pot. Indien er meedere spelers over zijn in deze fase, worden de kaarten in

de hand open gedraaid, en ontvangt de speler met de beste combinatie de pot.

Vervolgens schuift de dealer positie een plaats op. Ook de positie van de small blind en big blind

verplaatsen nu, en het volgende spel kan beginnen.

2.4 Behendigheidsniveau

Het behendigheidsniveau van een spel kan met de volgende formule worden uitgerekend (Borm en

Van der Genugten, 1998)1:

leereffect

Behendigheidsniveau =

leereffect + random effect

Het leereffect wordt in het kader van dit experiment weergegeven door het verschil in winst tussen

de ervaren speler en de beginner. Het verschil in winst tussen de fictieve speler en de ervaren speler

is het random effect, ofwel het toevalseffect. De ervaren speler heeft geen voorkennis over de

toevalselementen van het spel (hij weet niet welke kaarten de andere spelers hebben, en welke

kaarten er op tafel zullen komen te liggen). De fictieve speler is wel op de hoogte van deze

informatie. Samengevat:

Leereffect = winst ervaren speler - winst beginner,

Random effect = winst fictieve speler - winst ervaren speler.

1 Borm, P. and B. van der Genugten (1998). On the exploitation of casino games: how to

distinguish between games of chance and games of skill. In: F. Patrone, I. García-Jurado, and

S. Tijs (Eds.), Game Practice: Contributions from Applied Game Theory, pp. 19-33.

Dordrecht: Kluwer Academic Publishers.

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Kansspelen, waarbij het toevalseffect het leereffect domineert (zoals roulette), hebben een

behendigheidsniveau dat dicht bij 0 ligt. Behendigheidsspelen, waarbij het toevalseffect klein is in

vergelijking met het leereffect (zoals schaken), hebben een behendigheidsniveau dicht bij 1. Zoals

beargumenteerd in Hilbers et al. (2008)2 bedraagt het theoretische behendigheidsniveau van Texas

Hold’Em tenminste 0,30.

3. Opzet en uitvoering experiment

3.1 Doel experiment

Hoofddoel: experimenteel vaststellen van het behendigheidsniveau in Fixed Limit Texas Hold’Em, en

te zien of deze schatting in de buurt komt van het theoretische behendigheidsniveau van Texas

Hold’Em.

Nevendoel: inzicht te krijgen in de vorm van een beginnersstrategie.

3.2 Globale opzet experiment

Tijdens het experiment wordt Fixed Limit Texas Hold’Em gespeeld. Er wordt gespeeld aan drie tafels,

door zes spelers per tafel. In totaal zijn er dus 18 spelers nodig, hiervan zijn 16 spelers beginners, één

speler is een ervaren speler en één speler is een fictieve speler.

In totaal worden er 36 spellen gespeeld, verdeeld over drie spelrondes van 12 spellen. Na iedere

spelronde wordt er gepauzeerd, en wisselen de spelers van plaats.

Alle spelers ontvangen 500 fiches en deze fiches vertegenwoordigen echt geld. Aan het einde van het

experiment wisselen de spelers hun fiches in voor geld. De spelers hebben voldoende fiches om niet

voor het einde van de laatste spelronde blut te raken.

Na afloop wordt de precieze opzet en het doel van het experiment, alsmede een voorlopige schatting

van het behendigheidsniveau bekendgemaakt.

3.3 Details van de opzet van het experiment

Er worden 36 spellen per tafel gespeeld. In totaal zijn dit 108 spellen, de 108 benodigde kaartspellen

zijn voorafgaand aan het experiment voorbereid. Aan elk van de drie tafels worden dezelde spellen

synchroon gespeeld door andere spelers. Met behulp van de computer zijn de corresponderende

kaartvolgorden random gegenereerd. De kaartvolgordes van ieder spel aan de eerste tafel zijn dus

hetzelfde als de kaartvolgordes aan de andere tafels.

De spelers krijgen spelersnummers: 1,2,...,19. Speler 19 is een invalspeler, hij zal gedurende het

experiment invallen voor beginners die op een bepaald moment niet beschikbaar zijn. Speler 6 is de

2 Hilbers, P., R. Hendrickx, P. Borm and B. van der Genugten (2008). Skill in poker. Working

paper.

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fictieve speler, speler 12 de ervaren speler. Alle andere nummers worden door loting toegewezen

aan de beginners. Hiervan wordt een deelnemerslijst gemaakt.

De plaasten aan tafels zijn ook genummerd: bij tafel A (met de klok mee) de nummers A1,A2,...,A6,

soortgelijk bij tafel B en tafel C.

In de eerste fase krijgt de fictieve speler tafelpositie A6, de ervaren speler krijgt positie B6. De

overige posities worden door loting toegewezen, zo ook de belangrijke positie C6. De resultaten van

de beginner die op positie C6 zit worden namelijk vergeleken met de resultaten van de fictieve speler

en de ervaren speler.

In de tweede fase krijgt de fictieve speler tafelpositie B6, de ervaren speler tafelpositie C6, en de

belangrijke beginner (een andere speler dan in de eerste fase) positie A6.

In de derde fase krijgt de fictieve speler tafelpositie C6, de ervaren speler tafelpositie A6 en de

belangrijke beginner positie B6.

Het loten voor de verschillende posities geschiedt met de computer.

3.4 De voorbereidende fase

De 36 verschillende lotingsvolgorden worden met de computer gemaakt. De 108 kaartspellen

worden samengesteld als boven aangegeven.

De fictieve speler wordt aangezocht. Hij krijgt het eigenlijke doel van het experiment te horen. Hij

weet dat hij alleen beginners als tegenstander heeft. Voorts krijgt hij een lijst die voor elk van de 36

spellen alle kaartvolgorden geeft en die bovendien de (vaste) tafelposities vermeldt van de spelers

die aan het eind (na de river) respectievelijk een betere hand, een gelijke hand en een slechtere hand

gemaakt hebben dan hij.

De ervaren speler wordt aangezocht. Hij krijgt als onderdeel van het experiment te horen dat het

doel is na te gaan hoeveel beter een ervaren speler presteert dan een beginner. Hij weet dus ook dat

hij uitsluitend tegen beginners zal spelen. Hij krijgt niets over het bestaan van een fictieve speler te

horen.

De 17 beginners worden aangezocht. Hen wordt verteld dat ze niets hoeven voor te bereiden en dat

speluitleg op de dag zelve wordt gedaan, en dat het doel van het experiment het nagaan is wat het

effect is van een pokerface op het spelresultaat. Zij krijgen niets te horen over een ervaren speler en

een fictieve speler. Van de beginners wordt een beknopt profiel van hun pokerervaring gemaakt.

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3.5 De aanloopfase

De 19 spelers arriveren en krijgen eerst speluitleg (ook de ervaren en de fictieve speler zijn aanwezig

maar houden hun mond over alles wat met Texas Hold’Em te maken heeft, desgevraagd zijn ook zij

beginners).

Bij de speluitleg moet er wel een aantal oefenrondjes gespeeld worden om de spelregels te leren en

enigszins te verwerken, maar de instructeur moet er voor waken geen strategieadviezen te geven.

Dus beginners wordt wel duidelijk dat het fijn is met de beste hand in de showdown te zitten (dit is

het moment waarop de handkaarten open gaan, en de winnaar van de hand bekend wordt) en dat

met een slechte hand dit wel heel toevallig zou zijn. Er worden geen aanwijzingen gegeven over hoe

te handelen in specifieke situaties.

Vervolgens wordt verteld dat er in drie rondes gespeeld gaat worden. De spelers krijgen fiches en de

bijbehorende geldwaarde wordt bekend gemaakt. Daarna worden de spelernummers uitgeloot en

nemen de spelers hun eerste tafelposities in.

3.6 De speelfase

De eerste fase bestaat uit 12 spellen. De spelers ontvangen een spelerformulier, waarop ze voor

ieder spel van deze fase moeten aangeven welke beslissingen ze achtereenvolgens genomen hebben.

Voor ieder spel is een rij gereserveerd waarin de spelers de achtereenvolgens genomen beslissingen

noteren (deze beslissingen zijn: fold, call, raise, check en bet). Dit dient goed uitgelegd te worden.

Aan het einde van de eerste spelfase vermelden de spelers op hun formulier hoeveel fiches ze op dat

moment in bezit hebben, daarna worden de formulieren ingeleverd. Voor de tweede en de derde

spelronde ontvangen alle spelers een nieuw formulier en ze worden geacht hiermee op dezelfde

manier als in de eerste spelronde mee om te gaan.

Iedere tafel kent een spelleider (die zelf niet meespeelt). Deze deelt de kaarten (en mag in dit spel

dus nooit schudden of couperen). Hij ziet er op toe dat de spellen in de juiste volgorde worden

gespeeld. Verder controleert de spelleider dat de spelers hun formulieren invullen. Hij krijgt (ter

controle) een tafellijst: deze bevat 36 rijen voor elk spel. In iedere rij noteert hij ter controle de

riverkaart en de hoogte van de bereikte pot. Na afloop zijn er dus drie tafellijsten van de tafels A,B en

C beschikbaar. De spelleiders aan de drie tafels zorgen ervoor dat het spelen synchroon verloopt.

De spelers beginnen iedere spelronde met hetzelfde aantal fiches op tafel. Hiermee worden

psychologische effecten die een afhankelijkheid van de eerste fase en de tweede fase kunnen geven

uitgesloten. Tijdens het spelen mag geen commentaar uitgewisseld worden over te nemen of

genomen beslissingen.

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3.7 Tafelindeling

Tabel 5 geeft de tafelindeling voor iedere ronde weer. Deze indeling is tot stand gekomen door

middel van loting. De ervaren speler en de fictieve speler zitten iedere ronde op positie 6 (aan

verschillende tafels), zodat hun resultaten kunnen worden vergeleken. Ze krijgen immers precies

dezelfde kaarten. De beginners die op positie 6 zitten, zijn in dit experiment extra belangrijk. Ook hun

acties en resultaten worden vergeleken met die van de ervaren en fictieve speler. De belangrijke

beginners zijn speler 1 in ronde 1, speler 7 in ronde 2 en speler 10 in ronde 3.

Dealers Ronde 1 Ronde 2 Ronde 3

Tafel A Dealer 1 Dealer 1 Dealer 1

Tafel B Dealer 2 Dealer 2 Dealer 3

Tafel C Dealer 3 Dealer 3 Dealer 2

Spelers Ronde 1 Ronde 2 Ronde 3

A1 5 5 15

A2 16 2 17

A3 14 3 5

A4 18 17 13

A5 8 14 11

A6 6 7 12

B1 2 19 2

B2 15 1 19

B3 19 9 8

B4 9 8 18

B5 13 15 7

B6 12 6 10

C1 11 16 1

C2 4 4 16

C3 3 10 4

C4 7 11 9

C5 10 13 14

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C6 1 12 6

Tabel 5: Tafelindeling

3.8 Uitvoering

De invulformulieren die in dit experiment zijn gebruikt, bleken minder geschikt voor beginnende

spelers. Zodra een speler meedere acties in een fase deed (bijvoorbeeld eerst “check” en na een

raise van een speler achter hem “fold”), ontstonden er vaak problemen met het invullen. Meestal

werd uitsluitend de eerste actie genoteerd, slechts enkele spelers hebben meerdere acties per fase

ingevuld. Ook de dealers zijn niet overal zorgvuldig geweest. De grootte van de pot aan het einde van

ieder spel is vaak incorrect genoteerd, soms werd zelfs volstaan met een uitdrukking als “veel”, wat

niet erg handig is voor de analyse van de spellen. Het nagaan van wat er daadwerkelijk in ieder spel

gebeurd is, was nochtans mogelijk, maar vergde zodoende behoorlijk wat werk.

Voor een volgend experiment zijn de volgende zaken aan te bevelen:

- De invulformulieren dienen duidelijk te zijn voor spelers die nooit poker hebben gespeeld. - De dealers hebben de belangrijke taak om nauwkeurig to te zien dat ook de registratie

volgens de regels verloopt. - Het spelen van een proefronde, die niet meetelt voor het uiteindelijke resultaat is aan te

bevelen. Zo kan de ervaren speler wennen aan het spel van de beginners, en krijgen de

beginners handigheid in het correct noteren van hun acties.

Na afloop van het experiment heeft iedere speler zijn fiches ingeruild voor geld. Uit de op de

deelnemerslijsten vermelde fiches is een resultatenlijst gemaakt. Met iedere deelnemer

correspondeert een rij van drie getallen die zijn hoeveelheid fiches vermeldt aan het eind van iedere

fase. Deze lijst is in de computer ingevoerd en die heeft hieruit een schatting van het

behendigheidsniveau berekent. Vervolgens is de echte opzet van het experiment uitgelegd en is (met

de nodige reserve) de schatting bekendgemaakt.

4. Analyse experiment

4.1 Spelverlopen

De bijlage Spelverlopen bevat de acties van alle spelers in alle 36 spellen. Per spel kan hier bekeken

worden wat iedere speler in iedere fase gedaan heeft. Verder wordt weergegeven hoeveel fiches

iedere speler per spel heeft ingezet, wie de winnaar van het spel is, en wat de fichetotalen van de

spelers zijn na ieder spel. De eigen kaarten van de spelers, en ook de community cards, zijn hier terug

te vinden. Deze kunnen gebruikt worden om te bekijken hoe een typische beginner speelt: “met

welke handen verhoogt hij?”, “hoe speelt hij op de flop als hij een pair heeft gemaakt?”, etc.

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In deze bijlage valt ook te zien hoe actief de spelers zijn. De fictieve speler speelt erg weinig handen

(hij trekt zich vaak voor de flop terug). Dit is niet vreemd aangezien deze speler al voorafgaand aan

het spel al weet of hij de beste hand gaat maken of niet. De ervaren speler weet dat het inzetten met

marginale starthanden (zoals T7s, K3o) niet winstgevend is, ook deze speler speelt relatief weinig

handen. De beginners zijn een stuk actiever, de meeste van hen zijn bereid om met zowat iedere

hand (dus ook de slechtste handen in Texas Hold’Em, zoals 72o, 83o) fiches in te zetten om te zien of

de flop aansluiting biedt.

4.2 Resultatenlijst

Tabel 6 geeft de fichetotalen van alle spelers aan het einde van elk van de drie de spelronden.

Speler Ronde 1 Ronde 2 Ronde 3

1 540 483 374

2 553 530 531

3 447 461 461

4 518 499 464

5 570 566 577

6 534 695 914

7 449 479 476

8 456 428 475

9 444 457 542

10 495 539 507

11 551 554 527

12 467 557 558

13 473 407 470

14 456 427 484

15 507 418 356

16 515 463 246

17 500 512 526

18 469 469 488

19 556 556 524

Tabel 6: Fichetotalen van de spelers per ronde

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Hierbij dient te worden opgemerkt dat spelers 3,17 en 18 slechts in twee van de drie spelronden

hebben meegedaan. De grote winnaar in het experiment is de fictieve speler (speler 6). Hij heeft zijn

aantal fiches bijna weten te verdubbelen, de andere spelers komen hier niet bij in de buurt. In de

derde spelronde heeft hij een winst behaald van 219 fiches in slechts 12 spellen. De ervaren speler

(speler 12) heeft het experiment ook met winst afgesloten, maar komt toch niet in de buurt van de

fictieve speler. De eerste spelronde heeft hij zelfs met verlies af moeten sluiten. Zijn uiteindelijke

winst bedraagt 58 fiches. De drie beginners die op dezelfde posities als de ervaren en de fictieve

speler gespeeld hebben, hebben in de drie spelronden een winst van 38 fiches gerealiseerd. Tabel 7

geeft de winsten van de belangrijke spelers per spelronde.

Winst Belangrijke Beginner Ervaren Fictief

Ronde 1 40 -33 34

Ronde 2 30 90 161

Ronde 3 -32 1 219

Totaal 38 58 414

Tabel 7: Winsten van de belangrijke spelers per spelronde

In de uitzending van VARA-Nieuwslicht zijn de getallen gebruikt die voortkomen uit de ingevulde

resultatenlijst. Bovenstaande tabel komt voort uit de gecorrigeerde spelverlopen, en is dus niet

hetzelfde als de tabel die direct na afloop van het experiment is gemaakt. Gelukkig zijn de verschillen

klein. In het vervolg van dit verslag wordt vanzelfsprekend gerekend met de gecorrigeerde getallen.

4.3 Variantieanalyse resultaten

Met behulp van een beknopte variantieanalyse hebben we bepaald welke factoren (spelerstype,

spelronde, tafelposititie, etc.) het meest bepalend zijn voor de behaalde spelresultaten. Deze analyse

bevestigt het beeld dat de fictieve speler weliswaar een fiks hogere winst behaalt dan de andere

spelerstypes, maar dat het overall verschil tussen de ervaren speler en de beginners statistisch niet

significant is. Ook de overige factoren blijken statistisch nauwelijks van invloed te zijn. Het zeer

beperkt aantal van 36 spellen zorgt namelijk voor een hoge variantie in de resultaten.

4.4 Behendigheidsniveau

Het geschatte behendigheidsniveau van Texas Hold’Em op basis van alle rondes is 0,05. Dit komt

voort uit de volgende berekening (zie ook tabel 7):

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Leereffect = winst ervaren speler - winst beginner = 58 - 38 = 20

Random effect = winst fictieve speler - winst ervaren speler = 414 - 58 = 356

Behendigheidsniveau = leereffect / (leereffect + random effect) = 20/(20+356) = 0,05

De waarde van 0.05 is ver verwijderd van het theoretische behendigheidsniveau van minstens 0.30.

Hier zijn in het experiment ook wat duidelijke oorzaken voor te vinden. Door onregelmatigheden in

de gespeelde spellen is de winst van de ervaren speler aan het einde van het experiment erg laag. De

winst van de fictieve speler daarentegen is extreem hoog. Voor een meer realistische schatting van

het behendigheidsniveau zullen we een specifieke correctie doorvoeren aan de hand van

geconstateerde uitschieters.

In de eerste spelronde behaalt de belangrijke beginner een grotere winst dan de ervaren speler (die

zelfs verliest maakt). Dit resultaat is voornamelijk het gevolg van spel 9. In dit spel krijgen de spelers

op tafelpositie 6 (dus de ervaren en de fictieve speler, en de belangrijke beginner) de beste hand

gedeeld, namelijk AdAc. Tabel 8 toont dit spel, aan de tafel van de beginner.

Seat C1 C2 C3 C4 C5 C6

Speler 11 4 3 7 10 1

Hand Jc3c 6h3h AsJs JdJh 8d3s AdAc

Preflop SB BB Fold Bet

Fold Call Call Raise Raise

Call Call Call

Flop Check Bet Raise

Call Call Call

Turn Check Bet Raise

Call Call Call

River Check Check Bet

Fold Fold Raise Call

Tabel 8: Spel 9, Tafel C, Ronde 1

Speler 1 (de belangrijke beginner in de eerste spelronde), verhoogt tot twee keer toe preflop. Ook op

de flop (7s8c5d) en op de turn (Qh) is hij ervan overtuigd dat zijn hand de beste is, zelfs tegen drie

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tegenstanders. Hij blijft verhogen, ook op de river, en zijn hand blijkt bij de showdown ook inderdaad

de beste hand te zijn. Hij verdient 68 fiches met deze hand.

De ervaren speler ontvangt ook AdAc, dit spel verloopt volgens tabel 9.

Seat B1 B2 B3 B4 B5 B6

Speler 2 15 19 9 13 12



Fold Call Raise Raise Call

Call Call

Flop Bet Raise Raise

Call Call Call

Turn Bet Call Raise

Raise Call Fold Fold

River Bet

Fold

Tabel 9: Spel 9, Tafel B, Ronde 1

De acties op de preflop zijn vergelijkbaar met het spel aan tafel C, dezelfde vier spelers zien de flop.

De flop is 7s8c5d, niet bepaald een flop die speler 19 (op positie B3) gunstig gezind is. Met zoveel

raises op de preflop is het onwaarschijnlijk dat deze speler op de river de beste hand gaat hebben (hij

kan alleen nog winnen als de turn en de river allebei een schoppen brengen, dit heeft slechts een

kans van 5,49%). Dit weerhoudt hem echter niet om toch voor bet te kiezen, de volgende speler (met

JdJh, en dus wel een sterke hand) verhoogt, en de ervaren speler verhoogt nogmaals. Nog altijd met

vier spelers in het spel brengt de turn Qh. Waar de beginner zonder angst door blijft zetten, geeft de

ervaren speler de hand op. Hij denkt dat wanneer drie tegenstanders blijven verhogen, er toch op

zijn minst een speler is die zijn paar azen verslagen heeft. Als ook maar een van de tegenstanders een

van deze handen heeft: QQ, 88, 77, 55, 96, 64, dan is het voor hem correct om te passen. Tegen de

three of a kind handen kan de ervaren speler alleen nog winnen met een aas op de river (kans van

4,55%), tegen de handen die al een straight gemaakt hebben is de ervaren speler nu helemaal

kansloos (dit heet “drawing dead” in poker). Gezien de echte kaarten van de tegenstanders blijkt het

passen van de ervaren speler een dure misstap. Hij had op ieder moment de beste hand, en zou ook

in de showdown de pot gewonnen hebben. In plaats van ongeveer 75 fiches winst te maken in dit

spel, bereikt de ervaren speler een verlies van 22 fiches. Dit is dus een verschil van 97 fiches, wat een

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enorm aantal is. Dit spelresultaat valt met recht een uitschieter te noemen. Om nu te stellen dat de

ervaren speler een grote blunder heeft begaan, gaat toch wat te ver. Zijn inschatting (dat iemand

inmiddels een betere hand dan zijn AA gemaakt heeft) is logisch, maar helaas ook incorrect

(aangezien niemand ook maar in de buurt kwam van zijn hand). Dit spel heeft wel een erg grote

invloed op het behendigsheidsniveau in de eerste spelronde. Tabel 10 geeft de gecorrigeerde

winsten en het gecorrigeerde behendigheidsniveau.

Belangrijke Beginner Ervaren Fictief Behendigheidsniveau

Ronde 1 -28 -11 10 0,45

Tabel 10: Gecorrigeerde winsten en behendigheidsniveau in spelronde 1

In de tweede spelronde zijn geen uitschieters te vinden. In de derde spelronde gebeurt wel iets

opmerkelijks aan tafel C. Waar de beginners aan de andere tafels rustig en passief spelen (ze

verhogen zelden), schakelen sommige spelers aan tafel C nu over op een hyper-aggressieve speelstijl.

Zij verhogen wanneer dit mogelijk is, ongeacht de kracht van hun hand. Dit werkt natuurlijk sterk in

het voordeel van de fictieve speler, die in deze ronde een recordwinst boekt van 219 fiches. Het

toevalseffect is in deze ronde erg hoog, omdat de ervaren speler niet de kans kreeg om potten van

meer dan 100 fiches te kunnen winnen.

Het complete overzicht van behendigheidsniveaus, gecorrigeerd voor spel 9 in spelronde 1, in alle

spelrondes is weergegeven in tabel 11.

Belangrijke Beginner Ervaren Fictief Behendigheidsniveau

Ronde 1 -28 -11 10 0,45

Ronde 2 30 90 161 0,46

Ronde 3 -32 1 219 0,13

Totaal -30 80 390 0,26

Tabel 11: Gecorrigeerde winsten en behendigheidsniveau in alle rondes

Laten we spelronde 3 weg, dan krijgen we als behendigheidsniveau 0,46.

130

4.5 Beginnersstrategie

De meeste beginners in dit experiment bleken achteraf toch geen echte beginners te zijn; velen van

hen spelen regelmatig poker met vrienden, en sommigen spelen zelfs om echte dollars op internet.

Toch is een classificatie als beginner op basis van het spelgedrag bij veel van deze spelers wel correct,

gezien de inschattingsfouten die al deze spelers met regelmaat maken.

Iedere beginner speelt natuurlijk op een andere manier. Enkele globale conclusies wat betreft

spelgedrag zijn zeer wel mogelijk, zeker daar de speelstijl van de meeste beginners erg vaak

overeenkomt. In meerdere spelsituaties is het zelfs zo, dat vrijwel alle beginners dezelfde actie

ondernemen, terwijl de ervaren speler het juist anders aanpakt.

Ieder spel begint met het delen van de kaarten, en meteen op dit punt verschilt de strategie van de

ervaren speler enorm met de strategie van de meeste beginners. De openingsrange van de ervaren

speler (dit is de set van alle handen waarmee hij iets in gaat zetten) is afhankelijk van zijn positie aan

tafel. Hij speelt minder handen vanuit de positie direct na de big blind (deze positie heet “under the

gun”). Dit is een slechte positie, omdat er telkens een beslissing moet worden gemaakt, voordat alle

andere spelers hun actie bekendmaken. Het is onverstandig om vanuit deze positie met slechte

kaarten in de pot te komen. Vanuit betere posities (de dealer positie is de beste positie, deze beslist

namelijk altijd als laatste, en hier kunnen dus de overige acties worden meegenomen in de

beslissingen) speelt de ervaren speler meer handen. Typische minimale openingshanden voor de

slechte tafelpositie zijn {55, AQs, AKo}, d.w.z. dat deze speler verhoogt met pairs hoger dan 44, met

AQ en AK suited en met AK off-suit. Met alle overige handen wordt gepast.

De beginner let niet op tafelpositie. Voor hem is iedere positie hetzelfde, en hij baseert dus zijn

preflop actie puur op de sterkte van zijn hand, en eventueel op de acties die er al hebben

plaatsgevonden. Het grootste verschil met de ervaren speler ligt in het percentage van de handen

waarmee gespeeld wordt. Waar de ervaren speler zonder moeite een hand als A5o past, kiezen

vrijwel alle beginners voor de call optie. Ook met slechtere handen wordt vaak voor call gekozen;

sommige beginners callen zelfs met iedere hand. Tabel 12 toont het preflop gedrag van de spelers in

dit experiment.

Speler Ronde 1 Ronde 2 Ronde 3 %

1 9 5 10 66,67

2 3 4 4 30,56

3 11 8 - 79,17

4 11 6 12 80,56

5 7 10 9 72,22

6 5 7 5 47,22

7 8 5 7 55,56

131

8 10 7 12 80,56

9 11 12 10 91,67

10 10 7 9 72,22

11 8 9 11 77,78

12 4 4 4 33,33

13 9 9 6 66,67

14 11 9 3 63,89

15 9 9 10 77,78

16 11 10 12 91,67

17 - 4 9 54,17

18 11 - 11 91,67

19 7 8 8 63,89

Beginner 71,57

Ervaren 47,22

Fictief 33,33

Tabel 12: Flops gezien

Het percentage flops dat gehaald wordt door de beginners ligt ver boven het percentage van de

ervaren speler. De fictieve speler ziet nog minder flops, maar dat is logisch. Hij past namelijk over het

algemeen bij de preflop als hij weet dat iemand anders op de river een betere hand heeft dan hij. Het

gemiddelde over alle beginners ligt op 71,57% van alle handen. Gaan we ervan uit dat de beginners

de beste 71,57% van alle handen spelen, dan komt dat ongeveer neer op de range {22, 92s, T6o}.

Met alle suited handen beter dan 92s, en met alle handen beter dan T6o wordt dus gespeeld. Dit is

nogal een groot verschil met de range van de ervaren speler, die zelfs in de beste positie een hand als

K6s (ruim binnen de 71,57%) zal passen.

Het lijkt aantrekkelijk om met veel handen naar de flop te gaan, het kost meestal immers slechts 2

fiches, terwijl de meeste potten wel boven de 40 fiches uitkomen. Toch is dit op lange termijn geen

winstgevende strategie. Investeren met T6o loont dan wel als de flop 987 brengt, of T66, in de

meeste andere gevallen wordt gewoon 2 fiches aan een andere speler gedoneerd. Voor 2 fiches kans

maken op een grote pot (van 40 fiches) klinkt mooi, maar als de kans om die pot te winnen erg klein

is, en er meestal nog wel meer fiches geïnvesteerd dienen te worden, is dit een verliesgevende

tactiek.

132

Wat ook typisch is voor beginners in poker in het algemeen, maar ook voor de beginners in het

experiment is de voorkeur voor call boven bet/raise. De eerste gedachte van deze spelers is dat ze

proberen zo goedkoop mogelijk naar de river te komen, om aldaar te beslissen of ze de beste hand

hebben. Op de river wordt met goede handen wel veel verhoogd, daarvoor echter nogal weinig. We

nemen als voorbeeld spel 12 uit de eerste spelronde aan tafel C.

Spel 12

Community Cards 5s4s5dTd9d

Seat C1 C2 C5

Speler 11 4 10

Hand Qd4d KdTs Jh5h

Preflop BB Call Call

Check

Flop Check Check Check

Turn Bet Call Raise

Call Call


Raise Call Fold


De overige spelers zijn weggelaten. De speculatieve preflop call van speler 10 pakt deze keer goed

uit, hij maakt three of a kind al op de flop. De flop-actie is typisch voor een beginner. Speler 10 heeft

een ijzersterke hand na deze flop, het belangrijkste is nu om snel veel fiches in de pot te krijgen. Door

te checken zal dit echter niet lukken. Welk motief hij hiervoor ook heeft (hij is bang om de kracht van

zijn hand te verraden, of hij wil eerst de overige community cards zien), hier checken is onverstandig.

Op de flop heeft speler 10 maar liefst 91,81% kans om de pot te gaan winnen. Natuurlijk weet deze

speler niet wat de andere twee spelers op hand hebben, maar waarschijnlijk beseft hij wel dat zijn

hand nu het sterkste is. Als hij de andere spelers ertoe krijgt dat zij 2 fiches inzetten, dan zullen die in

91,81% van de gevallen aan het einde van het spel voor hem zijn. Dit is dus duidelijk een investering

met positieve verwachtingswaarde. Door te checken krijgt speler 11 gratis de kans om zijn hand te

verbeteren. Zijn winstkans gaat van 3,54% op de flop naar 16,67% op de turn, doordat de vierde

ruiten verschijnt. Speler 10 heeft nagelaten zijn sterke hand te beschermen door een bet te plaatsen

(wellicht was hij dan van speler 11 af geweest).

133

Een ander kenmerk in het spel van beginners is dat ze moeite hebben met passen. Dit kon al eerder

gezien worden op spel 9 toen de beginner met AA de pot won, waar de ervaren speler eieren voor

zijn geld koos en het spel op de turn verliet. Ook speler 3, die bij deze beginner aan tafel zat in dat

spel, kiest voor een opmerkelijke speelwijze, te zien in tabel 14.

Spel 9

Community Cards 7s8c5dQhQc

Speler 3 7 1

Hand AsJs JdJh AdAc

Preflop SB BB Bet

Call Raise Raise

Call Call


Call Call


Call Call


Fold Raise Call


Preflop is AJs een prima hand, al neemt de kracht van deze hand snel af na een raise en een reraise

door de tegenstanders (tegen AK en 99 bijvoorbeeld zal AJ slechts in 24% van de gevallen winnend

zijn). Na de flop, die niets brengt voor speler 3 zou hij toch eigenlijk de handdoek in de ring moeten

gooien. Hij speelt echter rustig verder. De flop kost hem 4 fiches, de turn zelfs 8. Een ervaren speler

had hier eenvoudig 12 fiches kunnen besparen.

Het laatste verschil tussen de tactiek van een ervaren speler en een beginner ligt in het bluffen.

Bluffen is een essentieel onderdeel van het pokerspel, en kan bij juiste uitvoering veel winst

134

opleveren. Belangrijk is de timing van een bluf. Een bluf is over het algemeen goed in twee specifieke

situaties. Het eerste geval doet zich voor als een speler op dit moment van het spel een waardeloze

hand heeft, maar met de juiste community cards toch een extreem sterke hand kan maken. Een

speler die bijvoorbeeld 8h5h heeft, en de getoonde community cards zijn Kh9h3s6c, heeft op dit

moment geen goede hand, maar met een harten op de river maakt hij waarschijnlijk de beste hand.

Een bet op de turn is nu in principe een bluf, de speler ziet het liefst alle andere spelers passen.

Mocht dit echter niet lukken, dan heeft hij nog steeds kansen om de pot te winnen. We noemen zo’n

bluf een semi-bluf, omdat als de bluf niet slaagt er nog niets verloren is.

Het andere geschikte moment om te bluffen is wanneer er nog weinig tegenstanders over zijn (liefst

slechts een), die geen van allen veel kracht tonen. Als alle speler zowel op de flop als op de turn

checken, valt het te proberen te pot te winnen door middel van een bluf.

De ervaren spelers kiest de momenten waarop hij een bluf plaatst zorgvuldig uit. De meeste

beginners doen dit niet, zij analyseren de situatie niet grondig genoeg, en kiezen vaak een willekeurig

moment uit om ervoor te gaan. Zij zijn zich niet bewust van de verhouding tussen de slagingskans

van een bluf en de kosten. Als voorbeeld tabel 15.

Spel 3

Community Cards Td8h6s3c3h

Seat C2 C3 C4

Speler 16 4 9

Hand Kc9c Kh7d Th6h

Preflop SB BB

Raise Call Raise

Call Call

Flop Check Bet

Raise Call Raise

Call Call

Turn Check Bet

Raise Call Raise

Raise Call Call

135

River Check Bet

Raise Fold Raise

Call


Speler 16 heeft blijkbaar besloten dat hij flink wil gaan bluffen in dit spel. Op een flop die voor hem

buiten een straight draw (met een 7 maakt speler 16 een straight) totaal niet geholpen heeft, en met

twee tegenstanders die redelijk wat kracht tonen, besluit hij zowel op de flop als op de turn te

reraisen. Zelfs op de river, waar zijn straight draw gemist heeft, probeert hij nogmaals de pot te

stelen.

Aan de hand van alle spelverlopen van dit experiment kan een karakterisering van de

beginnersstrategie in Texas Hold’Em gemaakt worden. Deze karakterisering kan mogelijk in de

toekomst worden gebruikt om het behendigheidsniveau van Texas Hold’Em uit te rekenen met

behulp van simulaties. Een typische beginner in dit experiment speelt als volgt:

- Preflop: De beginner speelt ruim 70% van zijn handen, wat neerkomt op minimum openingshanden van {22, 92s, T6o}. Met deze handen wordt voor call gekozen, ongeacht tafelpositie en ongeacht de acties die al door andere spelers zijn genomen. Zelfs met een hand als J8o wordt een raise en een reraise gecalld. De beginner raiset zelf nooit op de preflop, zelfs niet met AA.

- Flop en turn: Zolang een beginner nog kansen ziet om de beste hand te maken, blijft hij in het spel. Dit betekent dat hij met een flush-draw (nog 1 kaart nodig om een flush te maken), met een straight-draw (nog 1 kaart nodig voor een straight, bijvoorbeeld KT op een Q95 flop) en zelfs met overcards (de eigen kaarten van de beginner zijn hoger dan alle kaarten op de flop) een check/call strategie verkiest. Hij checkt dus wanneer dit nog kan, en callt anders de geplaatste bets. Ook met handen zwakker dan top pair (top pair houdt in dat er een pair is gemaakt met de hoogste kaart op de flop/turn, zoals K9 op een KJ4 flop) wordt gekozen voor check/call, ongeacht het aantal raises dat al is geplaatst. Zelfs met 22 op een QT4 flop wordt een reraise nog gecalld door de beginner. Met handen die op de flop/turn top pair of beter hebben gemaakt, wordt wel verhoogd, maar niet altijd. De beginner is bang om zijn sterke hand te verraden, en is ook bang voor ongunstige kaarten op de turn en river. Als niemand nog heeft verhoogd, zal met deze handen in de helft van de gevallen voor bet worden gekozen, en in de andere helft voor check. Als een andere speler al een bet heeft gedaan, kiest de beginner altijd voor call en nooit voor raise.

- River: Met alle pairs die slechter zijn dan top pair, kiest de beginner op de river voor een check/call tactiek. Hij is niet van plan zijn hand op de geven, maar wil wel zo goedkoop mogelijk zien of hij is verslagen. Bij de top pair handen is een bet/call strategie van toepassing, de beginner zal nooit folden, maar ook nooit raisen met alleen top pair. Met alle handen die beter zijn dan top pair wordt bet/raise gespeeld. Er wordt verhoogd zolang dit mogelijk is.

136

- Bluffen en sandbagging: Deze elementen maken geen onderdeel uit van de strategie van een beginner.

4.6 De ervaren speler

De ervaren speler heeft niet erg goed gepresteerd in dit experiment. De hand waarin hij zijn paar

azen opgaf bleek een dure vergissing. Die hand illustreert dat het moeilijk is om te spelen tegen

echte beginners. Sowieso brengt het spelen tegen beginners een grote variantie met zich mee.

Omdat er meer spelers in meer potten zitten, worden de potten groter, en zal de ervaren speler

vaker een spel verliezen. Als hij tegen goede spelers speelt en voor de flop, die 7h5s2c luidt,

gereraised heeft, kan hij er wel van uit gaan dat zijn paar koningen nog steeds goed is. Tegen

beginners zal hij aan zo’n tafel meer winnen per spel, maar ook af en toe verliezen van een beginner

die met 7d2d de raise van de ervaren speler gecalld heeft.

5. Conclusies

Doordat er slechts 36 spellen gespeeld zijn door iedere speler in dit experiment, is de schatting van

het behendigheidsniveau statistisch gezien niet erg betrouwbaar. Toch komt het geschatte

behendigheidsniveau van Texas Hold’Em goed in de buurt van de theoretische waarde (tenminste

0,30). Op basis van dit experiment is er dus geen aanleiding om de theoretische conclusie te herzien.

De (gecorrigeerde) behendigheid in de eerste twee spelrondes bedraagt respectievelijk 0,45 en 0,46.

In de derde spelronde komt het behendigheidsniveau uit op 0,13.

De ervaren speler had vooral in het begin moeite zich aan te passen aan de beginners. Hun strategie

bleek wezenlijk anders dan de strategie van de speler waar onze ervaren speler het normaal

gesproken tegen opneemt. Uit het spelgedrag van de beginners hebben we een globale

karakterisering van hun strategie kunnen destilleren. Deze kan voor een verdere precisering van het

theoretische behendigheidsniveau en voor toekomstige experimenten als input fungeren.

137

6. Bijlage: Spelverlopen

Spelronde 1:

Tafel A

Spel 1

Dealer Dealer 1

Community Cards 4d5d3s3dQh

Seat A1 A2 A3 A4 A5 A6

Speler 5 16 14 18 8 6

Hand Th8s Ks8h 9s2s Ah3c QcJc Ac5c

Preflop SB BB Call Call Call Call

Fold Check

1 2 2 2 2 2

Flop Check Bet Raise Call Fold

Fold Call

0 0 4 4 4 0

Turn Check Bet Fold

Fold

0 0 0 4 0 0

River 0 0 0 0 0 0

Total Invested 27 1 2 6 10 6 2

Stack Size 499 498 494 517 494 498

Spel 2

Community Cards 6h6cTc9hTd

138


Speler 5 16 14 18 8 6

Hand 7s6s Ts3h 9d2h Ac4h Qs3c Ad9s

Preflop SB BB Call Fold Fold

Call Call Check

2 2 2 2 0 0

Flop Bet Fold Fold

Raise Call

4 4 0 0 0 0

Turn Bet

Raise Call

8 8 0 0 0 0

River Bet

Raise Raise

Call

12 12 0 0 0 0


Stack Size 473 528 492 515 494 498

Spel 3

Community Cards Kd2h5dQs8s


Speler 5 16 14 18 8 6

Hand KsQh AcAh 3d3h 7d2d Ad3s 6d5s

Preflop SB BB Call Fold

Call Call Bet Fold Call

Call Call

139

4 4 4 2 4 0

Flop Fold Check

Check Check

0 0 0 0 0 0

Turn Check

Check Bet Call

Call

4 4 0 0 4 0

River Check

Bet Raise Fold

Call

8 8 0 0 0 0


Stack Size 503 512 488 513 486 498

Spel 4

Community Cards 4cKd4s7d6c


Speler 5 16 14 18 8 6

Hand Ac5s 5h2s Td9c Qc7s Kc7c Ah3s

Preflop SB BB Fold

Call Fold Call Call Check

2 0 2 2 2 0

Flop Check Bet

Fold Fold Call

0 0 0 2 2 0

Turn Bet Raise

140

Call

0 0 0 8 8 0

River Bet Call

0 0 0 4 4 0


Stack Size 501 512 486 497 506 498

Spel 5

Community Cards 3s6s3dAdKc


Speler 5 16 14 18 8 6

Hand Kh9s KsTd Ts4h Ac6h 7d4c 9h9c

Preflop SB BB

Fold Call Fold Call Call Check

0 2 0 2 2 2

Flop Check Check

Check Bet Call Fold

Call

0 2 0 2 2 0

Turn Check

Check Bet Call

Call

0 4 0 4 4 0

River Check

Check Bet Fold

Fold

0 0 4 4 0

141


Stack Size 501 504 486 515 498 496

Spel 6

Community Cards Ac5d7c3h8h


Speler 5 16 14 18 8 6

Hand 6h2c As4s KcTd Ts4h QcTc Jc4c

Preflop SB

BB Call Call Call Call Fold

Check

2 2 2 2 2 1

Flop Check Check Check Check Check

0 0 0 0 0 0

Turn Check Check Check Check Check

0 0 0 0 0 0

River Check Bet Fold Fold Fold

Fold

0 4 0 0 0 0


Stack Size 499 513 484 513 496 495

Spel 7

Community Cards 6d3sKsAd3d

142


Speler 5 16 14 18 8 6

Hand 9s6s KhTc Qh8d Ac5d 7h7s 3c2d


Call Check

2 2 2 2 2 2

Flop Check Check Check Check Check Check

0 0 0 0 0 0

Turn Check Check Check Check Check Bet

Fold Call Fold Fold Fold

0 4 0 0 0 4

River Check Bet

Call

0 4 0 0 0 4


Stack Size 497 503 482 511 494 513

Spel 8

Community Cards 5cQhKd4c9c


Speler 5 16 14 18 8 6

Hand Qc3h 7s6s 9s4s KhTs KcTc Jh5d

Preflop SB BB Call Call Fold

Fold Call Check

0 2 2 2 2 0

Flop Check Check Check Check

143

0 0 0 0 0 0

Turn Check Check Bet Call

Fold Fold

0 0 0 4 4 0

River Bet Raise

Call

0 0 0 8 8 0


Stack Size 497 501 480 497 512 513

Spel 9



Speler 5 16 14 18 8 6



Call Call Call Call

4 4 4 4 0 4

Flop Check Check Bet

Fold Fold Fold Raise Call

0 0 0 4 0 4

Turn Check Check

0 0 0 0 0 0

River Check Bet

Call

0 0 0 4 0 4

144


Stack Size 493 497 476 485 512 537

Spel 10

Community Cards 4c6dJc9s3c


Speler 5 16 14 18 8 6

Hand Jd5h 8h4s Qc8c 7s5c As4d Kd8s

Preflop SB BB Fold

Fold Call Call Call Check

0 2 2 2 2 0

Flop Check Check

Check Bet Raise Call

Fold Call

0 0 4 4 4 0

Turn Check Check

Check

0 0 0 0 0 0

River Check Check

Bet Call Call

0 0 4 4 4 0


Stack Size 493 495 498 475 502 537

Spel 11

Community Cards Jh3d2d9d5d

145


Speler 5 16 14 18 8 6

Hand 4c3s As4d 5h5c 9s2c Ah8c Qc5s

Preflop SB BB

Fold Call Call Call Call Check

0 2 2 2 2 2

Flop Check Check

Bet Call Fold Call Fold

0 2 2 0 2 0

Turn Bet

Call Call

0 4 4 0 4 0

River Bet

Call Fold

0 4 0 0 4 0


Stack Size 493 519 490 473 490 535

Spel 12



Speler 5 16 14 18 8 6

Hand Qd4d KdTs Jc5c Jd8c Jh5h 6c3h

Preflop SB

BB Call Bet Call Call Fold

Call Call

4 4 4 4 4 1

146

Flop Check Check Bet Fold Call

Raise Fold Raise Call

Call

6 0 6 0 6 0


Call Raise Call

Call

12 0 12 0 12 0

River Check Bet Call

Raise Raise Call

Call

12 0 12 0 12 0

Total Invested 111 34 4 34 4 34 1

Stack Size 570 515 456 469 456 534

Tafel B

Spel 1

Dealer Dealer 2



Speler 2 15 19 9 13 12


Preflop SB BB Fold Call Bet Fold

Call Call Call

4 4 0 4 4 0

Flop Check Check Bet Call

Fold Call

0 2 0 2 2 0

147

Turn Bet Raise Fold

Call

0 8 0 8 0 0

River Check Check

0 0 0 0 0 0


Stack Size 496 486 500 524 494 500

Spel 2



Speler 2 15 19 9 13 12


Preflop SB BB Call Fold Bet

Fold Fold Fold Call

0 1 2 4 0 4

Flop Check Bet

Call

0 0 0 2 0 2

Turn Check Bet

Fold

0 0 0 0 0 4

River

0 0 0 0 0 0


Stack Size 496 485 498 518 494 509

148

Spel 3



Speler 2 15 19 9 13 12


Preflop SB BB Fold Fold

Call Bet Call Call

Call

4 4 4 4 0 0

Flop Check Bet

Call Raise Call Raise

Call Call Call

6 6 6 6 0 0

Turn Check Bet

Raise Call Call Raise

Call Call Call

12 12 12 12 0 0

River Check Check

Check Check

0 0 0 0 0 0

Total Invested 88 22 22 22 22 0 0

Stack Size 562 463 476 496 494 509

Spel 4


149


Speler 2 15 19 9 13 12


Preflop SB BB Fold

Fold Fold Fold Call Check

0 0 0 2 2 0

Flop Check Bet

Fold

0 0 0 0 2 0

Turn

0 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 562 463 476 494 496 509

Spel 5



Speler 2 15 19 9 13 12


Preflop SB BB

Call Call Fold Bet Call Raise

Fold Call Call Call

0 6 0 6 6 6

Flop Check Bet

Fold Call Fold

150

0 0 0 2 0 2

Turn Bet

Call

0 0 0 4 0 4

River Bet

Call

0 0 0 4 0 4


Stack Size 560 457 476 524 490 493

Spel 6



Speler 2 15 19 9 13 12


Preflop SB

BB Call Call Call Bet Fold

Fold Call Call Call

2 4 4 4 4 1


Call Call

0 2 2 2 2 0

Turn Bet Call Call Call

151

0 4 4 4 4 0

River Check Check Bet Fold

Call Fold

0 4 0 4 0 0

Total Invested 51 2 14 10 14 10 1

Stack Size 558 494 466 510 480 492

Spel 7



Speler 2 15 19 9 13 12


Preflop SB BB Fold Call Bet Fold

Fold Call Call

1 4 0 4 4 0


Call Fold

0 2 0 0 2 0

Turn Check Bet

Raise Call

0 8 0 0 8 0

River Check Bet

Call

0 4 0 0 4 0


Stack Size 557 517 466 506 462 492

Spel 8

152



Speler 2 15 19 9 13 12



Call Call Check

2 2 2 2 2 0

Flop Check Check Check Bet

Fold Fold Fold Call

0 0 0 2 2 0

Turn Check Bet

Call

0 0 0 4 4 0

River Check Bet

Call

0 0 0 4 4 0


Stack Size 555 515 464 494 480 492

Spel 9



Speler 2 15 19 9 13 12



Fold Call Raise Raise Call

Call Call

153

0 8 8 8 0 8

Flop Bet Raise Raise

Call Call Call

0 6 6 6 0 6

Turn Bet Call Raise


0 12 12 4 0 8

River Bet

Fold

0 0 4 0 0 0

Total Invested 96 0 26 30 18 0 22

Stack Size 555 489 530 476 480 470

Spel 10



Speler 2 15 19 9 13 12


Preflop SB BB Fold

Fold Fold Call Call Check

0 0 2 2 2 0

Flop Check Check

Bet Call Fold

0 0 2 2 0 0

Turn Bet

Raise Call

0 0 8 8 0 0

154

River Bet

Raise Raise

Raise Call

0 0 16 16 0 0


Stack Size 555 489 560 448 478 470

Spel 11



Speler 2 15 19 9 13 12


Preflop SB BB

Fold Call Call Fold Call Check

0 2 2 0 2 2

Flop Check Check

Check Check

0 0 0 0 0 0

Turn Check Check

Bet Call Call Fold

0 4 4 0 4 0

River Bet

Call Call

0 4 4 0 4 0


Stack Size 555 511 550 448 468 468

155

Spel 12



Speler 2 15 19 9 13 12


Preflop SB

BB Call Call Bet Call Fold

Fold Call Call

2 4 4 4 4 1


Fold Call Fold

0 0 2 0 2 0

Turn Check Bet

Raise Raise

Call

0 0 12 0 12 0

River Check Bet

Call

0 0 4 0 4 0


Stack Size 553 507 556 444 473 467

Tafel C

Spel 1

Dealer Dealer 3


156


Speler 11 4 3 7 10 1



Call Check

2 2 2 2 2 2

Flop Check Check Check Bet Call Call

Fold Fold Fold

0 0 0 2 2 2

Turn Bet Fold Raise

Raise Call

0 0 0 8 0 8

River Bet Fold

0 0 0 4 0 0


Stack Size 498 498 498 522 496 488

Spel 2



Speler 11 4 3 7 10 1


Preflop SB BB Call Call Call

Call Call Check

2 2 2 2 2 2

Flop Bet Fold Fold Fold Fold

Call

157

2 2 0 0 0 0

Turn Check

Check

0 0 0 0 0 0

River Bet

Raise Call

8 8 0 0 0 0


Stack Size 486 518 496 520 494 486

Spel 3



Speler 11 4 3 7 10 1



Call Call Call Check

2 2 2 2 2 0


Check Bet Call Fold Fold

Call

2 2 2 0 0 0

Turn Check

Bet Raise Call

Raise Call Call

12 12 12 0 0 0

158

River Check

Bet Raise Fold

Raise Call

12 12 0 0 0 0


Stack Size 534 490 480 518 492 486

Spel 4



Speler 11 4 3 7 10 1


Preflop SB BB Call

Call Call Call Call Call

2 2 2 2 2 2


Check Check Check

0 0 0 0 0 0

Turn Bet Call Fold

Fold Fold Fold

0 0 0 4 4 0

River Bet Call

0 0 0 4 4 0


Stack Size 532 488 478 508 510 484

Spel 5


159


Speler 11 4 3 7 10 1


Preflop SB BB

Call Call Call Call Fold Check

2 2 2 2 1 2

Flop Bet

Fold Fold Fold Call

0 0 0 2 0 2

Turn Bet

Call

0 0 0 4 0 4

River Bet

Fold

0 0 0 0 0 4


Stack Size 530 486 476 500 509 499

Spel 6



Speler 11 4 3 7 10 1


160

Preflop SB

BB Call Call Fold Call Call

Call

2 2 2 0 2 2

Flop Bet

Fold Raise Fold Raise Call

Call

0 6 0 0 6 6

Turn Check

Check Check

0 0 0 0 0 0

River Check

Bet Call Fold

0 4 0 0 4 0


Stack Size 528 510 474 500 497 491

Spel 7



Speler 11 4 3 7 10 1


Preflop SB BB Call Bet Call Fold

Call Call Call

4 4 4 4 4 0

Flop Check Bet Fold Fold Call

Call

2 2 0 0 2 0

161

Turn Check Check Check

0 0 0 0 0 0

River Bet Call Fold

4 4 0 0 0 0


Stack Size 518 534 470 496 491 491

Spel 8



Speler 11 4 3 7 10 1



Fold Call Check

0 2 2 2 2 2

Flop Check Check Bet Raise

Fold Fold Call

0 0 0 4 4 0

Turn Bet Call

0 0 0 4 4 0

River Bet Raise

Call

0 0 0 8 8 0


Stack Size 518 532 468 478 515 489

162

Spel 9



Speler 11 4 3 7 10 1



Fold Call Call Raise Raise

Call Call Call

0 8 8 8 0 8


Call Call Call

0 4 4 4 0 4


Call Call Call

0 8 8 8 0 8



0 0 0 8 0 8

Total Invested 96 0 20 20 28 0 28

Stack Size 518 512 448 450 515 557

Spel 10



Speler 11 4 3 7 10 1


163

Preflop SB BB Bet

Fold Fold Call Fold Check

0 0 2 1 2 2

Flop Check Bet

Call Fold

0 0 2 0 0 2

Turn Bet

Call

0 0 4 0 0 4

River Bet

Raise Fold

0 0 8 0 0 4


Stack Size 518 512 463 449 513 545

Spel 11



Speler 11 4 3 7 10 1


Preflop SB BB

Fold Call Call Fold Call Bet

Call Call Call

0 4 4 0 4 4

Flop Check Check

Check Check

164

0 0 0 0 0 0

Turn Check Check

Bet Call Fold Fold

0 4 4 0 0 0

River Bet Raise

Call

0 8 8 0 0 0


Stack Size 518 536 447 449 509 541

Spel 12



Speler 11 4 3 7 10 1


Preflop SB

BB Call Fold Fold Call Fold

Check

2 2 0 0 2 1


0 0 0 0 0 0

Turn Bet Call Raise

Call Call

8 8 0 0 8 0


165

Raise Call Fold

8 8 0 0 4 0


Stack Size 551 518 447 449 495 540

Spelronde 2:

Tafel A

Spel 1

Dealer Dealer 1

Community Cards TcKsJhAs8h


Speler 5 2 3 17 14 7

Stack Size (Start) 570 553 447 500 456 449

Hand 4h2s Th6c Qs9s 7d2h AhJd 6d2c

Preflop SB BB Call Fold Call Fold

Call Call

2 2 2 0 2 0


0 0 0 0 0 0


Fold Fold

0 0 4 0 4 0

River Bet Call

0 0 4 0 4 0

166


Stack Size 568 551 461 500 446 449

Spel 2

Community Cards Ac8hAd3s2h


Speler 5 2 3 17 14 7

Hand 4c4s 9s3d Tc6d Qs2s 5s2d 7h7d

Preflop SB BB Fold Call Call

Call Fold Call

2 1 2 0 2 2



4 0 4 0 0 2

Turn Check

Bet Fold

4 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 581 550 455 500 444 445

Spel 3

Community Cards JsTd9hJc6s


167

Speler 5 2 3 17 14 7

Hand Ac4s 5c4d 3c2s 9s7s Kd3d KsQd

Preflop SB BB Call Call

Call Fold Fold Check

2 0 1 2 2 2


Call Call Call

2 0 0 2 2 2

Turn Check Check Bet

Fold Call Call

0 0 0 4 4 4


Fold Fold

0 0 0 0 0 4


Stack Size 577 550 454 492 436 466

Spel 4

Community Cards JdJhKs4cJs


Speler 5 2 3 17 14 7

Hand Tc3h Th8c Ts8d 8s5s Ad7c Jc8h

Preflop SB BB Fold

Fold Fold Call Fold Check

168

0 0 2 1 2 0

Flop Bet

Fold

0 0 0 0 2 0

Turn

0 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 577 550 452 491 439 466

Spel 5

Community Cards Jc7d8h2dJd


Speler 5 2 3 17 14 7

Hand Th5h 6c2h Ah3d Qh3h Ts4c AsKs

Preflop SB BB

Call Fold Call Fold Fold Bet

Call Call

4 0 4 0 1 4

Flop Bet

Raise Fold Call

4 0 0 0 0 4

Turn Bet

Raise Raise

169

Fold

8 0 0 0 0 12

River

0 0 0 0 0 0


Stack Size 561 550 448 491 438 487

Spel 6

Community Cards 9sQh6cQc6h


Speler 5 2 3 17 14 7

Hand AhTh 5h3s 8h2h 7c2d Td7d 7h4s

Preflop SB

BB Fold Fold Fold Call Fold

Call

2 0 0 0 2 1

Flop Check Bet

Fold

0 0 0 0 2 0

Turn

0 0 0 0 0 0

River

0 0 0 0 0 0

170


Stack Size 559 550 448 491 441 486

Spel 7

Community Cards 2sQsQd6hQc


Speler 5 2 3 17 14 7

Hand 9s4c Kh8d AsAc 7h3s 8c6c 7c4h


Fold Bet Raise Call

Call

1 6 6 0 6 0

Flop Check Check

0 0 0 0 0 0

Turn Check Bet

Call

0 4 4 0 0 0

River Check Bet

Call

0 4 4 0 0 0


Stack Size 558 536 469 491 435 486

Spel 8

Community Cards Kh7hQc5sJh


171

Speler 5 2 3 17 14 7

Hand 8c8d Ah2d Ts4s Kd5c 4h4c Kc4d

Preflop SB BB Fold Call Fold

Call Bet Call Call

Call

4 4 4 0 4 0

Flop Bet Call Fold

Raise Call Call

4 4 4 0 0 0

Turn Check Check

Bet Call Fold

4 4 0 0 0 0

River Check

Check

0 0 0 0 0 0


Stack Size 582 524 461 491 431 486

Spel 9

Community Cards 7c9d3h4s3s


Speler 5 2 3 17 14 7

Hand Qh5c 8s6d 7s2h Ts7h Td6h Ah6c

Preflop SB BB Fold Call


2 0 1 2 0 2

Flop Bet Fold

172

Call

2 0 0 2 0 0

Turn Check

Bet Call

4 0 0 4 0 0

River Check

Bet Call

4 0 0 4 0 0


Stack Size 570 524 460 506 431 484

Spel 10

Community Cards 9sKs7h4sQh


Speler 5 2 3 17 14 7

Hand Td8c Jc3d 6d4d Ah7d Jh9h Qc8d

Preflop SB BB Fold

Call Fold Call Call Check

2 0 2 2 2 0

Flop Check Check

Check Check

0 0 0 0 0 0

Turn Bet Fold

Fold Call

0 0 4 4 0 0

173

River Bet

Call

0 0 4 4 0 0


Stack Size 568 524 450 520 429 484

Spel 11

Community Cards 4s5s5hAhQh


Speler 5 2 3 17 14 7

Hand Js4c 7d6s 6d5d Th2c Td3s 9s8c

Preflop SB BB

Fold Call Fold Fold Call Call

0 2 0 0 2 2

Flop Check Bet

Raise Fold Fold

0 4 0 0 0 2

Turn

0 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 568 530 450 520 427 480

174

Spel 12

Community Cards AhQcKh8dAs


Speler 5 2 3 17 14 7

Hand Js4d Jd2s 5s5c 3d2h Tc9c 8c4h

Preflop SB

BB Fold Call Call Fold Fold

Check

2 0 2 2 0 1


Fold Call

0 0 2 2 0 0

Turn Check Bet

Call

0 0 4 4 0 0

River Check Check

0 0 0 0 0 0


Stack Size 566 530 461 512 427 479

Tafel B

Spel 1

Dealer Dealer 2


175


Speler 19 1 9 8 15 6

Stack Size (Start) 556 540 444 456 507 534


Preflop SB BB Bet Fold Raise Fold

Fold Call Call

1 6 6 0 6 0

Flop Check Bet Call

Fold

0 0 2 0 2 0

Turn Check Check

0 0 0 0 0 0

River Bet Call

0 0 4 0 4 0


Stack Size 555 534 463 456 495 534

Spel 2



Speler 19 1 9 8 15 6



Call Fold Call Call

4 1 4 4 0 4

176


Check

0 0 0 0 0 0

Turn Check Check Bet

Call Fold Fold

4 0 0 0 0 4

River Bet

Call

4 0 0 0 0 4


Stack Size 543 533 459 452 495 555

Spel 3



Speler 19 1 9 8 15 6


Preflop SB BB Call Bet

Call Fold Call Call Call

4 0 4 4 4 4


Call Fold Call Fold

2 0 0 2 0 2

Turn Check Bet

Fold Call

177

0 0 0 4 0 4

River Check Bet

Call

0 0 0 4 0 4


Stack Size 537 533 455 438 491 583

Spel 4



Speler 19 1 9 8 15 6


Preflop SB BB Bet

Fold Fold Call Call Call

0 0 4 4 4 4


Call Call Fold

0 0 2 2 0 2

Turn Check Bet

Fold Fold

0 0 0 0 0 4

River

0 0 0 0 0 0


Stack Size 537 533 449 432 487 599

178

Spel 5



Speler 19 1 9 8 15 6


Preflop SB BB

Call Fold Call Fold Call Bet

Call Call Call

4 0 4 0 4 4

Flop Check Bet

Fold Call Call

0 0 2 0 2 2

Turn Check Bet

Fold Raise Call

0 0 0 0 8 8

River Check Bet

Fold

0 0 0 0 0 4


Stack Size 533 533 443 432 473 623

Spel 6


179


Speler 19 1 9 8 15 6


Preflop SB

BB Fold Call Fold Call Bet

Call Call Call

4 0 4 0 4 4

Flop Check

Check Check Check

0 0 0 0 0 0

Turn Check

Bet Fold Fold Fold

4 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 545 533 439 432 469 619

Spel 7



Speler 19 1 9 8 15 6


Preflop SB BB Bet Fold Call Fold

Fold Raise Call Call

180

1 6 6 0 6 0

Flop Bet Raise Fold

Call

0 4 4 0 0 0

Turn Check Bet

Fold

0 0 4 0 0 0

River

0 0 0 0 0 0


Stack Size 544 523 456 432 463 619

Spel 8



Speler 19 1 9 8 15 6


Preflop SB BB Call Bet Fold

Call Call Call Call

4 4 4 4 4 0

Flop Bet Fold Fold Raise

Call Call

4 4 0 0 4 0

Turn Check Check

Bet Fold Call

4 0 0 0 4 0

181

River Check

Check

0 0 0 0 0 0


Stack Size 572 515 452 428 451 619

Spel 9



Speler 19 1 9 8 15 6



Call Bet Call Call

Call

4 4 4 4 0 0

Flop Check Check

Check Bet Call Call

Fold

0 2 2 2 0 0

Turn Check Check

Bet Call Call

0 4 4 4 0 0

River Check Bet

Fold Fold

0 0 0 4 0 0


Stack Size 568 505 442 452 451 619

182

Spel 10



Speler 19 1 9 8 15 6


Preflop SB BB Bet

Fold Fold Raise Call Call Call

0 0 6 6 6 6

Flop Bet Call Raise

Call Call Call

0 0 4 4 4 4

Turn Bet Raise Raise

Fold Call Call

0 0 0 12 12 12


Fold Call

0 0 0 0 4 4

Total Invested 84 0 0 10 22 26 26

Stack Size 568 505 432 430 425 677

Spel 11



Speler 19 1 9 8 15 6

183


Preflop SB BB

Call Bet Raise Fold Fold Fold

Call Call

6 6 6 0 1 2


Call Call

4 4 4 0 0 0


Fold Call

0 8 8 0 0 0

River Bet Call

0 4 4 0 0 0


Stack Size 558 483 467 430 424 675

Spel 12



Speler 19 1 9 8 15 6


Preflop SB

BB Fold Call Call Call Call

Check

2 0 2 2 2 2

Flop Check

Check Check Check Check

184

0 0 0 0 0 0

Turn Bet

Fold Call Fold Call

0 0 4 0 4 4

River Bet

Call Fold

0 0 4 0 0 4


Stack Size 556 483 457 428 418 695

Tafel C

Spel 1

Dealer Dealer 3



Speler 16 4 10 11 13 12

Stack Size (Start) 515 518 495 551 473 467


Preflop SB BB Call Fold Bet Fold

Fold Call Call

1 4 4 0 4 0


Call Call

0 2 2 0 2 0


185

Fold Call

0 0 8 0 8 0

River Check Bet

Raise Call

0 0 8 0 8 0


Stack Size 514 512 524 551 451 467

Spel 2



Speler 16 4 10 11 13 12



Call Fold Fold Call

4 1 2 4 0 4

Flop Check Bet

Call Fold

2 0 0 0 0 2

Turn Check

Bet Call

4 0 0 0 0 4

River Check

Bet Call

8 0 0 0 0 8


Stack Size 496 511 522 547 451 492

186

Spel 3



Speler 16 4 10 11 13 12



Call Bet Fold Call Call Call

Call

4 4 1 4 4 4


Raise Call Call Call Call

4 4 0 4 4 4

Check Check Bet

Turn Fold Fold Call Call

0 0 0 4 4 4


Call Fold

0 0 0 4 0 4


Stack Size 488 503 521 531 435 541

Spel 4


187


Speler 16 4 10 11 13 12


Preflop SB BB Fold

Call Call Fold Call Call

2 2 0 2 2 0

Flop Check Check

Check Check

0 0 0 0 0 0

Turn Check Check

Check Check

0 0 0 0 0 0

River Bet Fold

Fold Call

0 4 0 4 0 0


Stack Size 486 513 521 525 433 541

Spel 5



Speler 16 4 10 11 13 12


Preflop SB BB

Call Fold Call Call Call Bet

Call Call Call Call

4 0 4 4 4 4

Flop Check Check

188

Check Check Check

0 0 0 0 0 0

Turn Check Bet

Call Fold Fold Fold

4 0 0 0 0 4

River Check

Check

0 0 0 0 0 0


Stack Size 478 513 517 521 429 561

Spel 6



Speler 16 4 10 11 13 12


Preflop SB

BB Fold Fold Fold Call Fold

Bet Call

4 0 0 0 4 1

Flop Bet Call

2 0 0 0 2 0

Turn Bet Call

189

4 0 0 0 4 0

River Bet Fold

4 0 0 0 0 0


Stack Size 489 513 517 521 419 560

Spel 7



Speler 16 4 10 11 13 12


Preflop SB BB Bet Fold Fold Fold

Call Call

4 4 4 0 0 0


Fold Call

0 2 2 0 0 0

Turn Check Check

0 0 0 0 0 0

River Check Bet

Call

0 4 4 0 0 0


Stack Size 485 503 531 521 419 560

190

Spel 8



Speler 16 4 10 11 13 12



Call Call Check

2 2 2 2 2 0

Flop Check Check Bet Fold

Fold Fold Fold

0 0 0 2 0 0

Turn

0 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 483 501 529 529 417 560

Spel 9



Speler 16 4 10 11 13 12


191


Call Fold Fold Bet

Call

4 0 1 4 0 0

Flop Bet

Fold

0 0 0 2 0 0

Turn

0 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 479 501 528 534 417 560

Spel 10



Speler 16 4 10 11 13 12


Preflop SB BB Fold

Call Fold Call Bet Call

Call Call

4 0 4 4 4 0

Flop Check Check

Bet Fold Call Call

192

2 0 0 2 2 0

Turn Check Check

Bet Call Fold

4 0 0 4 0 0

River Check

Bet Call

4 0 0 4 0 0


Stack Size 465 501 524 558 411 560

Spel 11



Speler 16 4 10 11 13 12


Preflop SB BB

Fold Call Call Call Call Check

0 2 2 2 2 2

Flop Check Check

Check Check Check

0 0 0 0 0 0

Turn Check Check

Check Check Check

0 0 0 0 0 0

River Check Check

193

Check Check Check

0 0 0 0 0 0


Stack Size 465 499 532 556 409 558

Spel 12



Speler 16 4 10 11 13 12


Preflop SB

BB Fold Call Call Call Fold

Check

2 0 2 2 2 1


0 0 0 0 0 0

Turn Check Check Check Check

0 0 0 0 0 0

River Check Check Check Check

0 0 0 0 0 0


Stack Size 463 499 539 554 407 557

194

Spelronde 3:

Tafel A

Spel 1

Dealer Dealer 1

Community Cards QhQs7sKhKs


Speler 15 17 5 13 11 12

Stack Size (Start) 418 512 566 407 554 557

Hand JhTh 6c3h Qd4d 6h2c 7c5d Qc9d


Call Check

2 2 2 0 2 0


0 0 0 0 0 0


Fold Fold

0 0 4 0 4 0

River Bet Call

0 0 4 0 4 0


Stack Size 416 510 580 407 544 557

Spel 2

Community Cards Td3s3h6sKs

195


Speler 15 17 5 13 11 12

Hand QcTc 4c2s 8c7h 4h3c 4s2h Ah3d

Preflop SB BB Call Fold Fold

Bet Fold Call Call

4 1 4 4 0 0

Flop Check Check

Bet Fold Call

2 0 0 2 0 0

Turn Bet

Raise Call

8 0 0 8 0 0

River Bet

Call

4 0 0 4 0 0


Stack Size 398 509 576 430 544 557

Spel 3



Speler 15 17 5 13 11 12

Hand Qh5h Kc9c Kh7d Th6h Tc9s Qc7s

196


Call Call Call Check

2 2 2 2 2 0

Flop Check Bet Call

Fold Raise Fold Raise Call

Call

0 6 0 6 6 0

Turn Bet Call

Call

0 4 0 4 4 0

Bet Call

River Raise Call Call

0 8 0 8 8 0


Stack Size 396 489 574 474 524 557

Spel 4

Community Cards Ks7cQhJcTd


Speler 15 17 5 13 11 12

Hand Js8c AhTc Ad4c AcKh Kc3d Th2d

Preflop SB BB Fold

Call Call Call Bet Call

Call Call Call

4 4 4 4 4 0

197

Flop Bet Call

Fold Raise Call Call Call

0 4 4 4 4 0

Turn Bet Call

Raise Call Call Call

0 8 8 8 8 0

River Bet Fold

Raise Raise Raise

Call Call

0 16 16 16 0 0

Total Invested 116 4 32 32 32 16 0

Stack Size 392 496 580 481 508 557

Spel 5

Community Cards 8dAs4dAh9h


Speler 15 17 5 13 11 12

Hand 4h2d 5c2c KcQh 7s3d 6h5d AcJs

Preflop SB BB

Fold Call Call Fold Call Bet

Call Call Call

0 4 4 0 4 4

Flop Check Bet

Call Call Call

0 2 2 0 2 2

198

Turn Check Bet

Call Call Call

0 4 4 0 4 4

River Check Bet

Fold Call Fold

0 0 4 0 0 4

Total Invested 48 0 10 14 0 10 14

Stack Size 392 486 566 481 498 591

Spel 6

Community Cards 2c9sTh5s4h


Speler 15 17 5 13 11 12

Hand QhJh As8d 8h7h 3h2d 8s4c Tc4d

Preflop SB

BB Call Call Fold Call Fold

Bet Call Call Call

4 4 4 0 4 1


0 0 0 0 0 0

Turn Bet Fold Call Fold

199

4 0 4 0 0 0

River Bet Call

4 0 4 0 0 0


Stack Size 413 482 554 481 494 590

Spel 7

Community Cards Kd7dThQc4h


Speler 15 17 5 13 11 12

Hand 6h2c Ah8c Js6c 5h4s Jh3d 8s3h

Preflop SB BB Fold Call Call Fold

Call Check

2 2 0 2 2 0


0 0 0 0 0 0

Turn Check Check Check Check

0 0 0 0 0 0

River Check Check Check Bet

Fold Fold Fold

0 0 0 0 4 0


Stack Size 411 480 554 479 500 590

200

Spel 8

Community Cards 7h5d9d6h6d


Speler 15 17 5 13 11 12

Hand Jc9s KhKs 8s2d 7s5c Jh4s QhQd

Preflop SB BB Fold Call Bet

Raise Call Fold Fold Call Raise

Call Call Call

8 8 2 0 8 8

Flop Bet Fold Call

Raise Raise Call

Call

12 12 0 0 0 12

Turn Bet Fold

Call

4 4 0 0 0 0

River Bet

Call

4 4 0 0 0 0


Stack Size 383 538 552 479 492 570

Spel 9

Community Cards 2cKdQd4d6s


Speler 15 17 5 13 11 12

201

Hand 4s4h 7s3c 9h8c Th3s 8s5h Kc6c



2 0 1 2 2 0

Flop Check Check

Check

0 0 0 0 0 0

Turn Check Check

Bet Fold Fold

4 0 0 0 0 0

River

0 0 0 0 0 0


Stack Size 388 538 551 477 490 570

Spel 10

Community Cards 6s2sKs7hJh


Speler 15 17 5 13 11 12

Hand 8h6h Ad7d 7c5s 4s2h AcAh 5d2d

Preflop SB BB Fold

Call Call Call Fold Bet

Call Call Call

4 4 4 1 4 0

202

Flop Bet

Raise Fold Call Raise

Call Call

6 0 6 0 6 0

Turn Bet

Raise Call Call

8 0 8 0 8 0

River Bet

Call Fold

4 0 0 0 4 0


Stack Size 366 534 533 476 535 570

Spel 11

Community Cards Jd3d7s8cKd


Speler 15 17 5 13 11 12

Hand 5h3c 6c5s AcKc 9s3s 4c2s Kh2c

Preflop SB BB

Call Call Call Fold Call Check

2 2 2 0 2 2

Flop Check Check

Check Check Bet Fold Fold

Call Call

2 2 2 0 0 0

203

Turn Bet Call Call

4 4 4 0 0 0


Fold Fold

0 0 4 0 0 0


Stack Size 358 526 553 476 533 568

Spel 12

Community Cards Ac4hAd6h2c


Speler 15 17 5 13 11 12

Hand 8d3s 4s3c KhKd 9s7c 6c2d QcQs

Preflop SB

BB Fold Bet Call Call Raise

Fold Raise Call Call Call

2 0 6 6 6 6

Flop Check

Check Check Check

0 0 0 0 0 0

Turn Check

Check Check Check

0 0 0 0 0 0

River Check

Bet Fold Fold Call

204

0 0 4 0 0 4


Stack Size 356 526 577 470 527 558

Tafel B

Spel 1

Dealer Dealer 3



Speler 2 19 8 18 7 10

Stack Size (Start) 530 556 428 469 479 539


Preflop SB BB Call Bet Fold Call

Call Call Call

4 4 4 4 0 4

Flop Check Check Bet Fold Call

Fold Fold

0 0 2 0 0 2

Turn Bet Call

0 0 4 0 0 4

River Bet Call

0 0 4 0 0 4


Stack Size 526 552 434 465 479 545

205

Spel 2



Speler 2 19 8 18 7 10


Preflop SB BB Call Fold Call

Fold Fold Check

0 1 2 2 0 2

Flop Check Bet Fold

Call

0 0 2 2 0 0

Turn Check Bet

Call

0 0 4 4 0 0

River Check Bet

Call

0 0 4 4 0 0


Stack Size 526 551 422 480 479 543

Spel 3



206

Speler 2 19 8 18 7 10



Fold Call Call Check

0 2 2 2 2 2


Call Call Call

0 2 2 2 2 2

Check Bet Call Call

Turn Call Fold

0 4 0 4 4 4

River Bet Call Fold

Fold

0 0 0 4 4 0


Stack Size 526 543 418 512 467 535

Spel 4



Speler 2 19 8 18 7 10


Preflop SB BB Fold


207

0 2 2 2 2 0

Flop Bet Raise

Fold Call Raise Raise

Call Call

0 0 8 8 8 0

Turn Bet Call

Raise Call Call

0 0 8 8 8 0

River Bet Fold

Raise Raise

Raise Call

0 0 32 32 0 0

Total Invested 120 0 2 50 50 18 0

Stack Size 526 541 428 522 449 535

Spel 5



Speler 2 19 8 18 7 10


Preflop SB BB

Fold Fold Call Fold Call Call

0 0 2 0 2 2

Flop Bet Call

Call

208

0 0 2 0 2 2

Turn Bet Call

Call

0 0 4 0 4 4

River Check Check

Bet Fold Call

0 0 4 0 0 4


Stack Size 526 541 416 522 441 555

Spel 6



Speler 2 19 8 18 7 10


Preflop SB

BB Call Call Call Fold Call

Check

2 2 2 2 0 2

Flop Check

Check Check Bet Fold Call

Call Call

2 2 2 0 0 2

209

Turn Check

Check Check Bet Fold

Call Call

4 4 4 0 0 0


Fold Fold

0 0 4 0 0 0


Stack Size 518 533 438 520 441 551

Spel 7



Speler 2 19 8 18 7 10


Preflop SB BB Call Call Call Fold

Fold Check

1 2 2 2 2 0

Flop Bet Call Fold Fold

0 2 2 0 0 0

Turn Bet Call

0 4 4 0 0 0

River Bet Call

0 4 4 0 0 0


210

Stack Size 517 550 426 518 439 551

Spel 8



Speler 2 19 8 18 7 10



Fold Call Check

0 2 2 2 2 2

Flop Check Bet Call Fold Call

Call

0 2 2 2 0 2

Turn Check Bet Raise Fold

Fold Raise Call

0 0 8 8 0 0

River Bet Call

0 0 4 4 0 0


Stack Size 517 546 452 502 437 547

Spel 9


211


Speler 2 19 8 18 7 10



Call Fold Call Check

2 0 2 2 0 2


Bet Fold Fold Call

2 0 0 0 0 2

Turn Check

Bet Call

4 0 0 0 0 4

River Check

Bet Call

4 0 0 0 0 4


Stack Size 533 546 450 500 437 535

Spel 10


Seat

Speler 2 19 8 18 7 10


Preflop SB BB Fold


212

0 2 2 2 2 0

Flop Check Bet

Call Call Fold

0 2 2 0 2 0

Bet

Turn Call Raise Raise

Call Call

0 12 12 0 12 0

River Bet

Call Call

0 4 4 0 4 0


Stack Size 533 526 430 498 479 535

Spel 11



Speler 2 19 8 18 7 10


Preflop SB BB

Fold Fold Call Bet Fold Call

Call

0 0 4 4 1 4

Flop Check

Bet Call Call

213

0 0 4 4 0 4

Turn Check

Check Check

0 0 0 0 0 0

River Check

Bet Fold Call

0 0 4 0 0 4


Stack Size 533 526 451 490 478 523

Spel 12



Speler 2 19 8 18 7 10


Preflop SB

BB Call Call Call Call Call

Check

2 2 2 2 2 2

Flop Check

Check Check Bet Fold Fold Call

Fold Fold

0 0 2 0 0 2

Turn Check

Bet Raise

Call

0 0 8 0 0 8

214

River Check

Bet Call

0 0 4 0 0 4


Stack Size 531 524 475 488 476 507

Tafel C

Spel 1

Dealer Dealer 2



Speler 1 16 4 9 14 6

Stack Size (Start) 483 463 499 457 427 695


Preflop SB BB Call Fold Fold Bet


6 6 6 0 0 6

Flop Check Check Bet Raise


0 0 6 0 0 6

Turn Check Bet

Raise Call

0 0 12 0 0 12

River Bet Raise

Call

0 0 12 0 0 12


215

Stack Size 477 457 505 457 427 701

Spel 2



Speler 1 16 4 9 14 6



Raise Call Call Call Raise

Call Call Call Call

8 8 8 8 0 8


Raise Fold Fold Raise Raise

Call Call

8 0 0 8 0 8

Turn Bet Raise

Raise Raise Call

Call

16 0 0 16 0 16

River Bet Raise

Fold Raise Raise

Call

0 0 0 16 0 16

Total Invested 144 32 8 8 48 0 48

Stack Size 445 449 497 409 427 797

Spel 3


216


Speler 1 16 4 9 14 6



Bet Raise Call Raise

Call Call Call

6 6 6 6 0 0

Flop Check Bet

Fold Raise Call Raise

Call Call

0 6 6 6 0 0

Turn Check Bet

Raise Call Raise

Raise Call Call

0 16 16 16 0 0

River Check Bet

Raise Fold Raise

Call

0 12 0 12 0 0

Total Invested 114 6 40 28 40 0 0

Stack Size 439 409 469 483 427 797

Spel 4



Speler 1 16 4 9 14 6


217

Preflop SB BB Fold

Bet Raise Call Raise Call

Call Call Call

8 8 8 8 8 0

Flop Bet Call

Fold Raise Call Raise Call

Call Call

0 6 6 6 6 0

Turn Bet Call

Raise Call Raise Call


River 0 16 16 16 16 0

Bet Fold

Raise Raise Raise

Call Call Call

0 16 16 16 0 0

Total Invested 176 8 46 46 46 30 0

Stack Size 431 422 481 496 397 797

Spel 5



Speler 1 16 4 9 14 6


Preflop SB BB

Fold Bet Call Fold Call Raise

Call Call Call

0 6 6 0 6 6

218

Flop Check Bet

Raise Fold Fold Raise

Raise Call

0 8 0 0 0 8

Turn Bet

Raise Raise

Call

0 12 0 0 0 12

River Bet

Fold

0 0 0 0 0 4


Stack Size 431 396 475 496 391 835

Spel 6



Speler 1 16 4 9 14 6


Preflop SB

BB Call Call Call Fold Bet


Call Call Call Call

8 8 8 8 0 8

Flop Bet


219

Raise Call Call Call Call

8 8 8 8 0 8

Turn Bet

Raise Fold Call Fold Raise

Call Call

12 0 12 0 0 12

River Bet

Fold Fold

0 0 0 0 0 4

Total Invested 120 28 16 28 16 0 32

Stack Size 403 380 447 480 391 923

Spel 7



Speler 1 16 4 9 14 6


Preflop SB BB Call Call Fold Fold

Fold Check

1 2 2 2 0 0


0 0 0 0 0 0

Turn Check Check Check

0 0 0 0 0 0

River Check Check Check

220

0 0 0 0 0 0


Stack Size 402 378 452 478 391 923

Spel 8



Speler 1 16 4 9 14 6


Preflop SB BB Bet Fold Fold

Raise Raise Call Call

8 8 8 8 0 0

Flop Bet Call Raise


Call

8 8 8 8 0 0

Turn Bet Call Raise


Call

16 16 16 16 0 0


Raise Call Call

8 8 8 8 0 0

Total Invested 160 40 40 40 40 0 0

Stack Size 362 338 412 598 391 923

Spel 9

221



Speler 1 16 4 9 14 6



Bet Call Call Call Call

4 4 4 4 0 4

Flop Check Bet Call

Raise Fold Call Raise Fold

Call Call

6 0 6 6 0 2

Turn Check Bet

Raise Call Raise

Raise Call Call

16 0 16 16 0 0

River Check Bet

Raise Fold Fold

8 0 0 4 0 0

Total Invested 100 34 4 26 30 0 6

Stack Size 428 334 386 568 391 917

Spel 10



Speler 1 16 4 9 14 6


222

Preflop SB BB Fold

Bet Raise Call Call Raise

Call Call

8 8 8 8 8 0

Flop Check Bet

Fold Raise Call Call Raise

Call Call Call

0 6 6 6 6 0

Turn Check Bet

Raise Call Fold Raise

Raise Call Call

0 16 16 0 16 0

River Bet

Raise Fold Raise

Call

0 12 0 0 12 0

Total Invested 136 8 42 30 14 42 0

Stack Size 420 292 356 554 485 917

Spel 11



Speler 1 16 4 9 14 6


Preflop SB BB

Bet Call Call Call Fold Fold

4 4 4 4 1 2

223

Flop Bet Call Call Fold

2 2 2 0 0 0

Turn Bet Call Call

4 4 4 0 0 0


Fold Fold

0 0 4 0 0 0


Stack Size 410 282 383 550 484 915

Spel 12



Speler 1 16 4 9 14 6


Preflop SB

BB Bet Call Call Fold Fold


8 8 8 8 0 1

Flop Bet Raise Call Fold

Raise Raise Call

8 8 8 0 0 0

Turn Bet Raise Call

Raise Raise Call

224

16 16 16 0 0 0

River Bet Call Call

4 4 4 0 0 0

Total Invested 117 36 36 36 8 0 1

Stack Size 374 246 464 542 484 914

225

7. Bijlage: Resultaten per positie

Ronde 1

Tafel

Positie A B C

1 70 53 51

2 15 7 18

3 -44 56 -53

4 -31 -56 -51

5 -44 -27 -5

6 34 -33 40

Ronde 2

Tafel

Positie A B C

1 -4 0 -52

2 -23 -57 -19

3 14 13 44

4 12 -28 3

5 -29 -89 -66

6 30 161 90

Ronde 3

Tafel

Positie A B C

1 -62 1 -109

226

2 14 -32 -217

3 11 47 -35

4 63 19 85

5 -27 -3 57

6 1 -32 219

Documents

Tilburg University Texas Hold’em van der Genugten, B.B ... · Texas Hold’em: a game of skill Ben van der Genugten and Peter Borm CentER and Department of Econometrics and OR,