Game Theory: A brief introduction

Presented by Scott Corwon, IMPACTS

The purpose of the IMPACTS presentation series is to effectuateknowledge transfer concerning topical mathematical and scientificissues. Significant published works form the basis for much of thepresentation series, and these works are interpreted and presentedp p pby recognized leaders in the topic area. The presentation series ismade possible through the generous support of IMPACTS.

Game Theory

“ If it’s true that we are here to help others,then what exactly are the others here for? ”

- George Carlin

What is Game Theory?

Game Theory: The study of situations involving competing interests, modeled in terms of the strategies, probabilities, actions, gains, and losses of opposing players in a game. A general theory of strategic behavior with a common feature of Interdependence.

In other Words: The study of games to determine the probability of winning, given various strategies.

Example: Six people go to a restaurant.- Each person pays for their own meal – a simple decision problem- Before the meal, every person agrees to split the bill evenly among them – a game

A Little History on Game TheoryA Little History on Game TheoryJohn von Neumann and Oskar Morgenstern

Th f G d E B h- Theory of Games and Economic Behaviors

John Nash

- "Equilibrium points in N-Person Games", 1950, Proceedings of NAS.

"The Bargaining Problem", 1950, Econometrica.

"Non-Cooperative Games", 1951, Annals of Mathematics.

Howard W Kuhn Games with Imperfect informationHoward W. Kuhn – Games with Imperfect information

Reinhard Selten (1965) -“Sub-game Perfect Equilibrium" (SPE) (i.e. elimination by backward induction)

John C. Harsanyi - "Bayesian Nash Equilibrium"

Some Definitions for Understanding G thGame theory

Players-Participants of a given game or games.

Rules-Are the guidelines and restrictions of who can do what and when they can do it within a given game or games.

P ff i h f ili ( ll ) l i l ifi Payoff-is the amount of utility (usually money) a player wins or loses at a specific stage of a game.

Strategy- A strategy defines a set of moves or actions a player will follow in a given gy gy p y ggame. A strategy must be complete, defining an action in every contingency, including those that may not be attainable in equilibrium

Dominant Strategy -A strategy is dominant if, regardless of what any other players gy gy , g y p ydo, the strategy earns a player a larger payoff than any other. Hence, a strategy is dominant if it is always better than any other strategy, regardless of what opponents may do.

Important Review Questions for Game ThTheory

St tStrategyWho are the players? What strategies are available? What are the payoffs?

What are the Rules of the gamegWhat is the time-frame for decisions?What is the nature of the conflict?What is the nature of interaction?What information is available?

Five Assumptions Made to Understand Game TheoryGame Theory

1. Each decision maker ("PLAYER“) has available to him two or more well-specified choices or sequences of choices (called "PLAYS") choices or sequences of choices (called PLAYS ).

2. Every possible combination of plays available to the players leads to a well-defined end-state (win, loss, or draw) that terminates the game.

3. A specified payoff for each player is associated with each end-state (a ZERO-SUM game means that the sum of payoffs to all players is zero in each end-state).

4. Each decision maker has perfect knowledge of the game and of his opposition; that is, he knows in full detail the rules of the game as well as the payoffs of all other players.

5. All decision makers are rational; that is, each player, given two alternatives, will select the one that yields him the greater payoff.

Cooperative Vs Non CooperativeCooperative Vs. Non-CooperativeCooperative Game theory has perfect communication

d f fand perfect contract enforcement.

A ti i i hi h l A non-cooperative game is one in which players are unable to make enforceable contracts outside of those specifically modeled in the game. Hence, it is not defined p y gas games in which players do not cooperate, but as games in which any cooperation must be self-enforcing.

I t d d f Pl St t giInterdependence of Player Strategies1) Sequential – Here the players move in sequence, knowing

the other players’ previous moves.

-To look ahead and reason Back

2) S l H h l h 2) Simultaneous – Here the players act at the same time, not knowing the other players’ moves.

Use Nash Equilibrium to solve - Use Nash Equilibrium to solve

Simultaneous-move Games of Complete Information

A set of players (at least two players)A set of players (at least two players)S1 S2 SnS1 S2 SnS1 S2 ... SnS1 S2 ... Sn

For each player, a set of strategies/actionsFor each player, a set of strategies/actions

Payoffs received by each player for the combinations of the

p y , g /p y , g /{Player 1, {Player 1, S1, S1, Player 2,S2 ... Player SPlayer 2,S2 ... Player Snn}}

Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies

ui(s1, s2, ...sn), for all s1∈S1, s2∈S2, ... sn∈Sn

Nash’s Equilibrium Nash’s Equilibrium This equilibrium occurs when each player’s strategy is optimal, knowing q p y gy p , gthe strategy's of the other players.

A player’s best strategy is that strategy that maximizes that player’s payoff (utility), knowing the strategy's of the other players.

So when each player within a game follows their best strategy, a Nash equilibrium will occur.

Logic Logic

Definition: Nash Equilibrium

In the normal-form game {S1 , S2 , ..., Sn , u1 , u2 , ...,

Definition: Nash Equilibrium

In the normal form game {S1 , S2 , ..., Sn , u1 , u2 , ..., un}, a combination of strategies ),...,( **

1 nss is a Nash equilibrium if, for every player i,

***** Given others’

),...,,,,...,(

),...,,,,...,(**

1*

1*1

**1

**1

*1

niiii

niiii

sssssu

sssssu

+−

+−

≥

G e ot e schoices, player icannot be better-off if she deviates from s *

for all ii Ss ∈ . That is, *is solves

Maximize ),...,,,,...,( **1

*1

*1 niiii sssssu +−

si*

Subject to ii Ss ∈

Nash’s Equilibrium cont.:B i N h E ilib iB i N h E ilib i

The Nash Equilibrium of the imperfect-information game

Bayesian Nash EquilibriumBayesian Nash Equilibrium

A Bayesian Equilibrium is a set of strategies such that each player is playing a best response, given a particular set of beliefs about the move p y g p g pby nature.

All players have the same prior beliefs about the probability distribution p y p p yon nature’s moves.

– So for example, all players think the odds of player 1 being of a particular type is p, and the probability of her being the other type is 1-p

B ’ R l•• A mathematical rule of logic explaining how you should change A mathematical rule of logic explaining how you should change

b li f i li ht f i f ti b li f i li ht f i f ti

Bayes’ Rule

your beliefs in light of new information.your beliefs in light of new information.

•• BayesBayes’ Rule: ’ Rule: P(A|B) = P(B|A)*P(A)/P(B)P(A|B) = P(B|A)*P(A)/P(B)

•• To use To use BayesBayes’ Rule you need to know a few things:’ Rule you need to know a few things:•• To use To use BayesBayes Rule, you need to know a few things: Rule, you need to know a few things:–– You need to know You need to know PP((BB||AA))–– You also need to know the probabilities of You also need to know the probabilities of AA and and BB

Examples of Where Game Theo Can Be AppliedTheory Can Be Applied

Zero-Sum GamesPrisoner’s DilemmaNon-Dominant Strategy movesMixing MovesMixing MovesStrategic MovesBargainingg gConcealing and Revealing Information

Zero-Sum GamesPenny Matching:

Each of the two players has a penny.

Two players must simultaneously choose whether to show the Head or the Tail.

B h l k h f ll i lBoth players know the following rules:-If two pennies match (both heads or both tails) then player 2 wins player 1’s penny.

-Otherwise, player 1 wins player 2’s penny.

Player 2

--1 1 , , 11 1 1 , , --11

11 --11 --11 11Player 1

Tail

Head

Tail

Head

1 1 , , --11 --1 1 , , 11Tail

Prisoner’s DilemmaPrisoner’s DilemmaNo communication:

- Strategies must be undertaken without the full knowledge of what the other players (prisoners) will do.

Players (prisoners) develop dominant strategies but are not necessarily the best onenecessarily the best one.

Payoff Matrix for Prisoner’s DilemmaPayoff Matrix for Prisoner’s Dilemma

T dTedConfess Not Confess

ConfessBoth get 5 Both get 5

1 year for 1 year for Bill Bill

Bill

ggyearsyears 10 years for 10 years for

TedTed

Not Confess

10 years for 10 years for Bill Bill Both get 3Both get 3

1 year for 1 year for TedTed

ggyearsyears

Solving Prisoners’ DilemmaConfess is the dominant strategy for both Bill and Ted.

D d Dominated strategy-There exists another strategy which always does better regardless of other players’

choices-(Confess Confess) is a Nash equilibrium but is not always the best option-(Confess, Confess) is a Nash equilibrium but is not always the best option

Ted

fC f

PlayersStrategies

--5,5, --55 --11,,--1010

--1010 --11 --33 --33Bill

Not Confess

Confess

Not Confess

ConfessStrategies

--1010,,--11 --33,,--33Not Confess

Payoffs

Non Dominant strategy gamesNon-Dominant strategy gamesThere are many games when players do not have dominant strategiesy g p y g

- A player’s strategy will sometimes depend on the other player's strategy

According to the definition of Dominant strategy if a player depends - According to the definition of Dominant strategy, if a player depends on the other player’s strategy, he has no dominant strategy.

Non Dominant strategy gamesNon-Dominant strategy games

Ted

Confess Not Confess

7 years for Bill7 years for Bill 6 years for Bill 6 years for Bill

Confess

Bill2 years for Ted2 years for Ted 4 years for Ted4 years for Ted

Not Confess

9 years for Bill9 years for Bill

0 years for Ted0 years for Ted

5 years for Bill5 years for Bill

3 years for Ted3 years for Ted

Solution to Non-Dominant st ateg gamesstrategy games

Ted Confesses Ted doesn’t confessBill BillBill Bill

Confesses Not confess Confesses Not confess

7 years 9 years 6 years 5 years

Best Strategies

There is not always a dominant strategy and sometimes your best strategy will depend on the other players move.

Examples of Where Game Theo Can Be AppliedTheory Can Be Applied

Mixing MovesMixing MovesExamples in Sports Examples in Sports (Football & Tennis)(Football & Tennis)

Strategic MovesStrategic MovesStrategic MovesStrategic MovesWar War ––Cortes Burning His Own ShipsCortes Burning His Own Ships

BargainingBargainingBargainingBargainingSplitting a PieSplitting a Pie

Concealing and Revealing InformationConcealing and Revealing InformationBluffing in PokerBluffing in Poker

Applying Game Theory to NFLApplying Game Theory to NFLSolving a problem within the Salary Cap.

How should each team allocate their Salary cap. (Which position should get more money than the other)

The Best strategy is the most effective allocation of the team’s money to obtain the most wins.

Correlation can be used to find the best way to allocate the team’s money.

What is a correlation?What is a correlation?

A correlation examines the relationship between two measured pvariables.- No manipulation by the experimenter/just observed. - E.g., Look at relationship between height and weight.

You can correlate any two variables as long as they are numerical ( i l i bl )(no nominal variables)

Is there a relationship between the height and weight of the students in this room?in this room?- Of course! Taller students tend to weigh more.

Salaries vs Points scored/AllowedSalaries vs. Points scored/AllowedPositionPosition CorrelatioCorrelatio

nnTT--testtest Position Position CorrelatioCorrelatio

n n TT--testtest

RBRB .27.27 2.672.67

kk .25.25 2.522.52

TETE 1717 1 741 74

DEDE .25.25 2.522.52

CBCB .15.15 1.481.48

SS 0606 6161TETE .17.17 1.741.74

OLOL .04.04 .34.34

QBQB .03.03 .32.32

SS .06.06 .61.61

LBLB .05.05 .52.52

DTDT .04.04 .34.34

WRWR --.03.03 --.30.30 PP 00 00

Running Backs edge out Kickers for best correlation of position Running Backs edge out Kickers for best correlation of position spending to team points scored. Tight Ends also show some modest relationship between spending and points.

The Defensive Linemen are the top salary correlators with The Defensive Linemen are the top salary correlators, with cornerbacks in the second spot

Total Position spending vs WinsTotal Position spending vs. WinsPosition Wins Points Scored Points Allowed Total Position

Correlation Correlation Correlation Spending

K 0.27 0.27 0.17 0.27

CB 0.17 0.12 0.12 0.23

TE 0.16 0.2 0.15 0.17

OL 0.15 0.02 0.2 0.08

RB 0.11 0.11 -0.03 0.26

QB 0.1 0.08 0.08 0.04

DE 0.08 -0.14 0.17 0.16

P 0.08 0.01 0.03 0.04

LB 0.05 -0.08 0.15 -0.02

S 0.03 0.02 0.05 0.04

DT -0.02 -0.01 0.02 -0.04

WR 0 08 0 01 0 04 0 01WR -0.08 -0.01 -0.04 0.01

Note: Kicker has highest correlation also OL is ranked high alsoNote: Kicker has highest correlation also OL is ranked high also.

What this meansWhat this meansNFL teams are not very successful at delivering results for the big money spent on individual players.

Th ' h h k l b There's high risk in general, but more so at some positions over others in spending large chunks of your salary cap space.

Future StudyFuture StudyIncrease the Sample size.

Cluster Analysis

Correspondence analysis

Exploratory Factor Analysis

ConclusionConclusionThere are many advances to this theory to help describe and prescribe the right strategies in many different situations.

Although the theory is not complete, it has helped and will continue to help many people in solving strategic gamescontinue to help many people, in solving strategic games.

ReferencesReferencesNasar, Sylvia (1998), A Beautiful Mind: A Biography of John Forbes Nash, Jr., Winner of the Nobel Prize in Economics, 1994. Simon and Schuster, New York.

Rasmusen, Eric (2001), Games and Information: An Introduction to Game Theory, 3rd ed. Blackwell, Oxford.

Gibbons, Robert (1992), Game Theory for Applied Economists. Princeton University Press, Princeton NJPrinceton, NJ.

Mehlmann, Alexander. The Games Afoot! Game Theory in Myth and Paradox. AMS, 2000.

Wiens, Elmer G. Reduction of Games Using Dominant Strategies.Vancouver: UBC M.Sc. Thesis, 1969.

H. Scott Bierman and Luis Fernandez (1993) Game Theory with Economic Applications, 2nd ed. (1998), Addison-Wesley Publishing Co.

D. Blackwell and M. A. Girshick (1954) Theory of Games and Statistical Decisions, John Wiley & S N Y kSons, New York.

NFL Official, 2004 NFL Record and Fact Book; Time Inc. Home Entertainment, New York, New York.

Game Theory

“ If it’s true that we are here to help others,then what exactly are the others here for? ”

- George Carlin