Dominant and Dominated Strategieshrtdmrt2/Teaching/GT_2016_19/L2.pdfDominant and Dominated Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign

Dominant and Dominated Strategies

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

Junel 8th, 2016

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 The Extensive Form Representation of a Game

2 Strategies and the Normal Form Representation of a Game

3 Randomized Choices

4 Exercises

5 Formalizing the Game

6 Dominant and Dominated Strategies

7 Iterated Delation of Strictly Dominated Strategies

8 Iterated Delation of Dominated Strategies

9 Exercises


The Extensive Form Representation of a Game

On the Agenda




4 Exercises





9 Exercises



What is a Game?

I From the noncooperative point of view, a game is a multi-person decision situationdefined by its structure, which includes:

- Players: Independent decision makers

- Rules: Which specify the order of players’ decisions, their feasible decisionsat each point they are called upon to make one, and the information theyhave at such points.

- Outcome: How players’ decisions jointly determine the physical outcome.

- Preferences: players’ preferences over outcomes.

C. Hurtado (UIUC - Economics) Game Theory 1 / 39


Examples

I Matching Pennies (version A).

Players: There are two players, denoted 1 and 2.

Rules: Each player simultaneously puts a penny down, either heads up or tails up.

Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.

I Matching Pennies (version B).


Rules: Player 1 puts a penny down, either heads up or tails up. Then, Player 2puts a penny down, either heads up or tails up.




Examples

I Matching Pennies (version C).


Rules: Player 1 puts a penny down, either heads up or tails up, without lettingplayer 2 know his decision. Player 2 puts a penny down, either heads up or tails up.


I Matching Pennies (version D).


Rules: Players flip a fair coin to decide who begins. The looser puts a penny down,either heads up or tails up. Then, the winner puts a penny down, either heads upor tails up.

Outcomes: If the two pennies match, the looser pays 1 dollar to player 2;otherwise, the winner pays 1 dollar to player 1.




I Some games that are important in economics have simultaneous moves.

I ”Simultaneous” means strategically simultaneous, in the sense that players’decisions are made without knowledge of others’ decisions.

I It need not mean literal synchronicity, although that is sufficient for strategicsimultaneity.

I But many important games have at least some sequential decisions, with somelater decisions made with knowledge of others’ earlier decisions.

I We need a way to describe and analyze both kinds of game.

I One way to describe either kind of game is via the extensive form or game tree,which shows a game’s sequence of decisions, information, outcomes, and payoffs.




I A version of Matching Pennies with sequential decisions, in which Player 1 movesfirst and player 2 observes 1’s decision before 2 chooses his decision.

I




I We can represent the usual Matching Pennies with simultaneous decisions byintroducing an information set, which includes the decision nodes a player cannotdistinguish and at which he must therefore make the same decision, as in thecircled nodes.

I




I The order in which simultaneous decision nodes are listed has some flexibility, as inprevious case, where player 2 could have been at the top.

I For sequential decisions the order must respect the timing of information flows.(Information about decisions already made, as opposed to predictions of futuredecisions, has no reverse gear.)

I All decision nodes in an information set must belong to the same player and havethe same set of feasible decisions. (Why?)

I Players are normally assumed necessarily to have perfect recall of their own pastdecisions (and other information). If so, the tree must reflect this.

DefinitionA game is one of perfect information if each information set contains a single decisionnode. Otherwhise, it is a game of imperfect information.



The Extensive Form Representation of a GameI This is an example of a game with simultaneous decision nodes and players with

perfect recall of their own past decisions.

IC. Hurtado (UIUC - Economics) Game Theory 8 / 39



I This is an example of a game with simultaneous decision nodes and playerswithout perfect recall of their own past decisions.

I




I This is another example of a game with simultaneous decision nodes and playerswithout perfect recall of their own past decisions.

I




I Shared uncertainty (in economics ”symmetric information”) can be modeled byintroducing moves by an artificial player (without preferences) called Nature, whochooses the structure of the game randomly, with commonly known probabilities.

I


Strategies and the Normal Form Representation of a Game

On the Agenda




4 Exercises





9 Exercises




I For sequential games it is important to distinguish strategies from decisions oractions.

I A strategy is a complete contingent plan for playing the game, which specifies afeasible decision for each of a player’s information sets in the game.

I Recall that his decision must be the same for each decision node in an informationset.

I A strategy is like a detailed manual of actions, not like a single decision or action.




I




I It is assumed that conditional on what a player observes, he can predict theprobability distributions of his own and others’ future decisions and theirconsequences.

I If players have this kind of foresight, then their rational sequential decision-makingin ”real time” should yield exactly the same distribution of decisions assimultaneous choice of fully contingent strategies at the start of play.

I The player writes his own manual of actions. Then he will give you (a neutralreferee) the manual and let you play out the game. You will tell him who won.

I Because strategies are complete contingent plans, players must be thought of aschoosing them simultaneously (without observing others’ strategies),independently, and irrevocably at the start of play.




I Why a strategy must be a complete contingent plan, specifying decisions even fora player’s own nodes that he knows will be ruled out by his own earlier decisions?

I Otherwise, other players’ strategies would not contain enough information for aplayer to evaluate the consequences of his own alternative strategies.

I We would then be unable to correctly formalize the idea that a strategy choice isrational.

I Putting the point in an only seemingly different way, in individual decision theory,zero probability events can be ignored as irrelevant, at least for expected-utilitymaximizers.

I But in games zero-probability events cannot be ignored because what has zeroprobability is endogenously determined by players’ strategies.




I

Player 2 strategies:I Strategy 1 (s1): Play H if player 1 plays H; Play H if player 1 plays TI Strategy 2 (s2): Play H if player 1 plays H; Play T if player 1 plays TI Strategy 3 (s3): Play T if player 1 plays H; Play H if player 1 plays TI Strategy 4 (s4): Play T if player 1 plays H; Play T if player 1 plays T




I A game maps strategy profiles (one for each player) into payoffs (with outcomesimplicit).

I A game form maps strategy profiles into outcomes, without specifying payoffs.

I Specifying strategies make it possible to describe an extensive-form game’srelationship between strategy profiles and payoffs by its (unique) normal form orpayoff matrix or (usually when strategies are continuously variable) payoff function.




I

I



Strategies and the Normal Form Representation of a GameI The mapping from the normal to the extensive form isn’t univalent: the normal

form for Matching Pennies version B has possible extensive forms other than theone depicted before:

I


Randomized Choices

On the Agenda




4 Exercises





9 Exercises


Randomized Choices

Randomized Choices

I In game theory it is useful to extend the idea of strategy from the unrandomized(pure) notion we have considered to allow mixed strategies (randomized strategychoices).

I Example: Matching Pennies Version A has no appealing pure strategies, but thereis a convincingly appealing way to play using mixed strategies: randomizing 50-50.(Why?)

I Our definitions apply to mixed as well as pure strategies, given that theuncertainty about outcomes that mixed strategies cause is handled (just as forother kinds of uncertainty) by assigning payoffs to outcomes so that rationalplayers maximize their expected payoffs.

I Mixed strategies will enable us to show that (reasonably well-behaved) gamesalways have rational strategy combinations.

I In extensive-form games with perfect recall, mixed strategies are equivalent tobehavior strategies, probability distributions over pure decisions at each node(Kuhn’s Theorem; see MWG problem 7.E.1).


Exercises

On the Agenda




4 Exercises





9 Exercises


Exercises

Exercises

I Exercise 1. In a game where player i has N information sets indexed n = 1, · · · ,Nand Mn possible actions at information set n, how many strategies does player ihave?

I Exercise 2. Depict the normal formm of Matching Pennies Version C.

I


Exercises

Exercises

I Exercise 3. Consider the followign two-player (excluding payoffs):

I

a) What are player 1’s possible strategies? player 2’s?

b) Suppose that we change the game by merging the information set of player 1’ssecond round of moves (so that all the four nodes are now in a single informationset). Argue why the game is no longer one of perfect recall.


Formalizing the Game

On the Agenda




4 Exercises





9 Exercises




I Up to this point we defined game without been formal. Let me introduce someNotation:

- set of players: I = {1, 2, · · · ,N}

- set of actions: ∀i ∈ I, ai ∈ Ai , where each player i has a set of actions Ai .

- strategies for each player: ∀i ∈ I, si ∈ Si , where each player i has a set ofpure strategies Si available to him. A strategy is a complete contingent planfor playing the game, which specifies a feasible action of a player’sinformation sets in the game.

- profile of pure strategies: s = (s1, s2, · · · , sN) ∈∏N

i=1 Si .

Note: let s−i = (s1, s2, · · · , si−1, si+1, · · · , sN) ∈ S−i , we will denotes = (si , s−i ) ∈ (Si , S−i ).

- Payoff function: ui :∏N

i=1 Si → R, denoted by ui (si , s−i )




I Now we can denote game with pure strategies and complete information in normalform by: ΓN = {I, {Si}i , {ui}i}.

I What about the games with mix strategies?

I We have taken it that when a player acts at any information set, hedeterministically picks an action from the set of available actions. But there is nofundamental reason why this has to be case.

DefinitionA mixed strategy for player i is a function σi : Si → [0, 1], which assigns a probabilityσi (si ) ≥ 0 to each pure strategy si ∈ Si , satisfying

∑si∈Si

σi (si ) = 1.

I We denote the set of mixed strategies by ∆(Si ).

I Note that a pure strategy can be viewed as a special case of a mixed strategy inwhich the probability distribution over the elements of Si is degenerate.



Example

I Meeting in New York:

- Players: Two players, 1 and 2

- Rules: The two players can not communicate. They are suppose to meet in NYCat noon to have lunch but they have not specify where. Each must decide whereto go (only one choice).

- Outcomes: If they meet each other, they enjoy other’s company. Otherwise, theyeat alone.

- Payoffs: They attach a monetary value of 100 USD to other’s company and 0USD to eat alone.

I

player 2A B C

player 1A 100,100 0,0 0,0B 0,0 100,100 0,0



Example

I Meeting in New York:

- set of players: I = {1, 2}

- set of actions: A1 = {A,B}, and A2 = {A,B,C}

- strategies for each player: S1 = A1, and S2 = A2 (Why?)

- Payoff function: ui :∏2

i=1 Si → R, denoted by ui (si , s−i )

u(si , s−i ) ={

1000

if si = s−i

if si 6= s−i

I Player 2

- pure strategies: S2 = {A,B,C}. Player 2 has 3 pure strategies.

- mixed strategies:∆(S2) = {(σ2

1 , σ22 , σ

23) ∈ R3|σ2

m ≥ 0∀m = 1, 2, 3 and∑3

m=1 σ2m = 1}



On the Agenda




4 Exercises





9 Exercises




I Now we turn to the central question of game theory: What should be expected toobserve in a game played by rational agents who are fully knowledgeable about thestructure of the game and each others’ rationality?

I To keep matters simple we initially ignore the possibility that players mightrandomize in their strategy choices.

I The prisoner’s dilemma:* Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary

confinement with no means of speaking to or exchanging messages with the other.* The prosecutors do not have enough evidence to convict the pair on the principal charge.

They hope to get both sentenced to a year in prison on a lesser charge.* Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the

opportunity either to: betray the other by testifying that the other committed the crime,or to cooperate with the other by remaining silent.

* Here is the offer:- If A and B each betray the other, each of them serves 2 years in prison- If A betrays B but B remains silent, A will be set free and B will serve 3 years in

prison (and vice versa)- If A and B both remain silent, both of them will only serve 1 year in prison (on the

lesser charge)




I Let me put prisoner’s dilemma as a game of trust:

I

player 2trust cheat

player 1trust 5,5 1,10cheat 10,1 2,2

I Observe that regardless of what her opponent does, player i is strictly better offplaying Cheat rather than Trust. This is precisely what is meant by a strictlydominant strategy.

I Player 2 plays Trust. Player 1 knows that 10 > 5, better to Cheat.I Player 2 plays Cheat. Player 1 knows that 2 > 1, better to Cheat.I Regardless of the other’s strategies, it is always better to Cheat.I Note that both would be better off if they both play trust.I Lesson: self-interested behavior in games may not lead to socially optimal

outcomes.C. Hurtado (UIUC - Economics) Game Theory 28 / 39



DefinitionA strategy si ∈ Si is a strictly dominant strategy for player i if for alls̃i 6= si and all s−i ∈ S−i , ui(si , s−i) > ui(s̃i , s−i).

I A strictly dominant strategy for i uniquely maximizes her payoff for any strategyprofile of all other players.

I If such a strategy exists, it is highly reasonable to expect a player to play it. In asense, this is a consequence of a player’s ”rationality”.




I What about if a strictly dominant strategy doesn’t exist?

I

player 2a b c

player 1A 5,5 0,10 3,4B 3,0 2,2 4,5

I You can easily convince yourself that there are no strictly dominant strategies herefor either player.

I Notice that regardless of whether Player 1 plays A or B, Player 2 does strictlybetter by playing b rather than a.

I That is, a is ”strictly dominated” by b.




DefinitionA strategy si ∈ Si is strictly dominated for player i if there exists astrategy s̃i ∈ Si such that for all s−i ∈ S−i , ui(s̃i , s−i) > ui(si , s−i). In thiscase, we say that s̃i strictly dominates si .

I In words, s̃i strictly dominates si if it yields a strictly higher payoff regardless ofwhat (pure) strategy rivals use.

I Note that the definition would also permits us to use mixed strategies

I Using this terminology, we can restate the definition of strictly dominant: Astrategy si is strictly dominant if it strictly dominates all other strategies.

I It is reasonable that a player will not play a strictly dominated strategy, aconsequence of rationality, again.


Iterated Delation of Strictly Dominated Strategies

On the Agenda




4 Exercises





9 Exercises




I

player 2a b c

player 1A 5,5 0,10 3,4B 3,0 2,2 4,5

I We argued that a is strictly dominated (by b) for Player 2; hence rationality ofPlayer 2 dictates she won’t play it.

I We can push the logic further: if Player 1 knows that Player 2 is rational, heshould realize that Player 2 will not play strategy a.

I Notice that we are now moving from the rationality of each player to the mutualknowledge of each player’s rationality.

I Once Player 1 realizes that 2 will not play a and ”deletes” this strategy from thestrategy space, then strategy A becomes strictly dominated by strategy B forPlayer 2.

I If we iterate the knowledge of rationality once again, then Player 2 realizes that 1will not play A, and hence ”deletes” A.

I Player 2 should play c. We have arrived at a ”solution”.




DefinitionA game is strict-dominance solvable if iterated deletion of strictlydominated strategies results in a unique strategy profile.

I Since in principle we might have to iterate numerous times in order to solve astrict-dominance solvable game, the process can effectively can only be justified bycommon knowledge of rationality.

I As with strictly dominant strategies, it is also true that most games are notstrict-dominance solvable.

I You might worry whether the order in which we delete strategies iterativelymatters. Insofar as we are working with strictly dominated strategies so far, it doesnot.


Iterated Delation of Dominated Strategies

On the Agenda




4 Exercises





9 Exercises




DefinitionA strategy si ∈ Si is a weakly dominant strategy for player i if for alls̃i 6= si and all s−i ∈ S−i , ui(si , s−i) ≥ ui(s̃i , s−i), and for at least onechoice of s−i the inequality is strict.

DefinitionA strategy si ∈ Si is weakly dominated for player i if there exists a strategys̃i ∈ Si such that for all s−i ∈ S−i , ui(s̃i , s−i) ≥ ui(si , s−i), and for at leastone choice of s−i the inequality is strict. In this case, we say that s̃i weaklydominates si .

DefinitionA game is weakly-dominance solvable if iterated deletion of weaklydominated strategies results in a unique strategy profile.




I Using this terminology, we can restate the definition of weakly dominant: Astrategy si is weakly dominant if it weakly dominates all other strategies.

I You might worry whether the order in which we delete strategies iterativelymatters. Delation of dominated strategies could leave to different outcomes.

I

P2L R

U 5,1 4,0P1 M 6,0 3,1

D 6,4 4,4

P2L R

P1U 5,1 4,0D 6,4 4,4

P2L R

P1M 6,0 3,1D 6,4 4,4


Exercises

On the Agenda




4 Exercises





9 Exercises


Exercises

ExercisesI Exercise 1. Prove that a player can have at most one strictly dominant strategy.I Exercise 2. Apply the iterated elimination of strictly dominated strategies to the

following normal form games. Note that in some cases there may remain morethat one strategy for each player. Say exactly in what order you eliminated rowsand columns.

I Exercise 3. Apply the iterated elimination of dominated strategies to the followingnormal form games. Note that in some cases there may remain more that onestrategy for each player. Say exactly in what order you eliminated rows andcolumns.


Exercises

Exercises

I Exercise 2 (cont.).

I


Exercises

ExercisesI Exercise 2 (cont.).

I


Exercises

Exercises

I Exercise 2 (cont.).

I


Documents

Dominant and Dominated Strategieshrtdmrt2/Teaching/GT_2016_19/L2.pdfDominant and Dominated Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign