Game TheoryGame Theory

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

- George Carlin- George Carlin

What is Game Theory?What is Game Theory? Game TheoryGame Theory: The study of situations involving : The study of situations involving

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

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

Example:Example: Six people go to a restaurant. Six people go to a restaurant.-- Each person pays for their own meal – Each person pays for their own meal – a simple decision a simple decision

problemproblem-- Before the meal, every person agrees to split the bill Before the meal, every person agrees to split the bill evenly among them – evenly among them – a gamea game

A Little History on Game TheoryA Little History on Game Theory John von Neumann and Oskar Morgenstern John von Neumann and Oskar Morgenstern

- - Theory of Games and Economic BehaviorsTheory of Games and Economic Behaviors John NashJohn Nash

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

"The Bargaining Problem", 1950, "The Bargaining Problem", 1950, EconometricaEconometrica. .

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

Howard W. Kuhn – Games with Imperfect informationHoward W. Kuhn – Games with Imperfect information Reinhard Selten (1965) -“Sub-game Perfect Equilibrium" Reinhard Selten (1965) -“Sub-game Perfect Equilibrium"

(SPE) (i.e. elimination by backward induction) (SPE) (i.e. elimination by backward induction) John C. Harsanyi - "Bayesian Nash Equilibrium" John C. Harsanyi - "Bayesian Nash Equilibrium"

Some Definitions for Understanding Some Definitions for Understanding Game theoryGame theory

PlayersPlayers-Participants of a given game or games. -Participants of a given game or games.

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

PayoffPayoff-is the amount of utility (usually money) a player wins or -is the amount of utility (usually money) a player wins or loses at a specific stage of a game.loses at a specific stage of a game.

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

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

Important Review Questions for Important Review Questions for Game TheoryGame Theory

StrategyStrategy• Who are the players? Who are the players? • What strategies are available? What strategies are available? • What are the payoffs?What are the payoffs?

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

Five Assumptions Made to Five Assumptions Made to Understand Game TheoryUnderstand Game Theory

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

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

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

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

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

Cooperative Vs. Non-CooperativeCooperative Vs. Non-Cooperative

Cooperative Game theory has perfect Cooperative Game theory has perfect communication and perfect contract communication and perfect contract enforcement.enforcement.

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

Interdependence of Player Interdependence of Player StrategiesStrategies

1)1) Sequential – Here the players move Sequential – Here the players move in sequence, knowing the other in sequence, knowing the other players’ previous moves.players’ previous moves.

- To look ahead and reason Back- To look ahead and reason Back

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

- Use Nash Equilibrium to solve - Use Nash Equilibrium to solve

Simultaneous-move Games of Complete Simultaneous-move Games of Complete InformationInformation

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

uuii((ss11, s, s22, ...s, ...snn), ), for allfor all ss11SS11, s, s22SS22, ... s, ... snnSSnn

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

For each player, a set of strategies/actionsFor each player, a set of strategies/actions{Player 1, {Player 1, S1, S1, Player 2,S2 ... Player SPlayer 2,S2 ... Player Snn}}

Nash’s Equilibrium Nash’s Equilibrium

This equilibrium occurs when each player’s strategy This equilibrium occurs when each player’s strategy is optimal, knowing the strategy's of the other is optimal, knowing the strategy's of the other players.players.

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

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

Logic Logic

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,

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

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

**1

*1

*1

**1

**1

*1

niiii

niiii

sssssu

sssssu

for all ii Ss . That is, *is solves

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

*1

*1 niiii sssssu

Subject to ii Ss

Definition: Nash EquilibriumDefinition: Nash Equilibrium

Given others’ choices, player i cannot be better-off if she deviates from si*

Nash’sNash’s Equilibrium cont.:Equilibrium cont.: Bayesian Nash EquilibriumBayesian Nash Equilibrium

The Nash Equilibrium of the imperfect-The Nash Equilibrium of the imperfect-information gameinformation game

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

• All players have the same prior beliefs about the All players have the same prior beliefs about the probability distribution on nature’s moves.probability distribution on nature’s moves.

– So for example, all players think the odds of player 1 So for example, all players think the odds of player 1 being of a particular type is being of a particular type is pp, and the probability of her , and the probability of her being the other type is being the other type is 1-p1-p

• A mathematical rule of logic explaining A mathematical rule of logic explaining how you should change your beliefs in how you should change your beliefs in light of new information.light of new information.

• Bayes’ Rule: Bayes’ Rule:

P(A|B) = P(B|A)*P(A)/P(B)P(A|B) = P(B|A)*P(A)/P(B)

• To use Bayes’ Rule, you need to know a few To use Bayes’ Rule, you need to know a few things: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

Bayes’ RuleBayes’ Rule

Examples of Where Game Examples of Where Game Theory Can Be AppliedTheory Can Be Applied

Zero-Sum GamesZero-Sum Games Prisoner’s DilemmaPrisoner’s Dilemma Non-Dominant Strategy movesNon-Dominant Strategy moves Mixing MovesMixing Moves Strategic MovesStrategic Moves BargainingBargaining Concealing and Revealing Concealing and Revealing

InformationInformation

Zero-Sum GamesZero-Sum Games Penny Matching:Penny Matching:

Each of the two players has a penny. Each of the two players has a penny.

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

Both players know the following rules:Both players know the following rules:-If two pennies match (both heads or both tails) then player -If two pennies match (both heads or both tails) then player

2 wins player 1’s penny. 2 wins player 1’s penny.

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

Player 1

Player 2

Tail

Head

Tail

Head

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

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

Prisoner’s DilemmaPrisoner’s Dilemma

No communication:No communication:

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

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

Payoff Matrix for Prisoner’s Payoff Matrix for Prisoner’s DilemmaDilemma

ConfessConfess

BillBill

Not Not ConfessConfess

TedTedConfessConfess Not Confess Not Confess

Both get 5 Both get 5 yearsyears

1 year for 1 year for Bill Bill

10 years for 10 years for TedTed

10 years for 10 years for Bill Bill

1 year for 1 year for TedTed

Both get 3Both get 3

yearsyears

Solving Prisoners’ DilemmaSolving Prisoners’ Dilemma Confess is the dominant strategy for both Bill and Confess is the dominant strategy for both Bill and

Ted.Ted.

Dominated strategyDominated strategy-There exists another strategy which always does better -There exists another strategy which always does better

regardless of other players’ choicesregardless of other players’ choices-(Confess, Confess) is a Nash equilibrium but is not always -(Confess, Confess) is a Nash equilibrium but is not always

the best optionthe best option

-5,-5, -5 -5 -1-1,-10,-10

-10-10,-1,-1 -3-3,-3,-3

Bill

Ted

Not Confess

Confess

Not Confess

Confess

Players

Strategies

Payoffs

Non-Dominant strategy gamesNon-Dominant strategy games

There are many games when players do There are many games when players do not have dominant strategiesnot have dominant strategies

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

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

Non-Dominant strategy gamesNon-Dominant strategy games

ConfessConfess

BillBill

Not Not ConfessConfess

TedTed

ConfessConfess Not Confess Not Confess

7 years for Bill7 years for Bill

2 years for Ted2 years for Ted

6 years for Bill 6 years for Bill

4 years for Ted4 years for Ted

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 Solution to Non-Dominant strategy gamesstrategy games

Ted ConfessesTed Confesses Ted doesn’t confessTed doesn’t confess BillBill BillBill

Confesses Not confessConfesses Not confess ConfessesConfesses Not confessNot confess

7 years7 years 9 years 6 years 9 years 6 years 5 years 5 years

Best StrategiesBest Strategies

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

Examples of Where Game Examples of Where Game Theory Can Be AppliedTheory Can Be Applied

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

Strategic MovesStrategic MovesWar –Cortes Burning His Own ShipsWar –Cortes Burning His Own Ships

BargainingBargainingSplitting a PieSplitting a Pie

Concealing and Revealing InformationConcealing and Revealing InformationBluffing in PokerBluffing in Poker

Applying Game Theory to NFLApplying Game Theory to NFL

Solving a problem within the Salary Cap.Solving a problem within the Salary Cap. How should each team allocate their How should each team allocate their

Salary cap. (Which position should get Salary cap. (Which position should get more money than the other)more money than the other)

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

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

What is a correlation?What is a correlation?

A correlation examines the A correlation examines the relationshiprelationship between two measured variables.between two measured variables.

- No manipulation by the experimenter/just - No manipulation by the experimenter/just observed. observed.

- E.g., Look at relationship between height and - E.g., Look at relationship between height and weight.weight.

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

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

Salaries vs. Points scored/AllowedSalaries vs. Points scored/Allowed

PositionPosition CorrelationCorrelation T-testT-test

RBRB .27.27 2.672.67

kk .25.25 2.522.52

TETE .17.17 1.741.74

OLOL .04.04 .34.34

QBQB .03.03 .32.32

WRWR -.03-.03 -.30-.30

Position Position Correlation Correlation T-testT-test

DEDE .25.25 2.522.52

CBCB .15.15 1.481.48

SS .06.06 .61.61

LBLB .05.05 .52.52

DTDT .04.04 .34.34

PP 00 00

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 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 0.01

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

What this meansWhat this means

NFL teams are not very successful at NFL teams are not very successful at delivering results for the big money delivering results for the big money spent on individual players. spent on individual players.

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

Future StudyFuture Study

Increase the Sample size.Increase the Sample size. Cluster AnalysisCluster Analysis Correspondence analysisCorrespondence analysis Exploratory Factor Analysis Exploratory Factor Analysis

ConclusionConclusion

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

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

ReferencesReferences Nasar, 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, NJ.

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

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

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

Questions?Questions?

Comments?Comments?