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Game Theory

By Ben Cutting &

Rohit

Venkat

Game Theory: General Definition

Mathematical decision making tool

Used to analyze a competitive situation in order to determine the optimal course of action

Involves at least two players who usually must choose an action from at least two options

A player’s payoff (what they gain/lose from the game) is determined by both their own choice and the choices of other players

Players act “rationally” in their decision

making, try to maximize their payoff

History

John von Neumann published a series of papers in 1928 pertaining to game theory

Theory of Games and Economic Behavior

by John von Neumann and Oskar Morgenstern (1944)

Initially developed to analyze competitions in which one individual does better at another’s expense (zero sum games)

Developed extensively in the 1950s by many scholars to treat a wide class of interactions

Key Terms

Nash Equilibrium

–

state in which each player has a given strategy that provides them with their maximum payoff. Therefore no player has an incentive to change their strategy unilaterally

Strategy

–

a player’s plan of action that accounts for all possible game scenarios.

Completely describes

a player’s behavior

Representations

Two classical representations: matrix form and tree formMatrix form is traditionally associated with simultaneous move games

1, 1 3, 4 4, 2 -1, -1

Player A

1

2

1

2

Player B

Representations (cont.)

Tree formOutcomes often change if the type of game is changed

Player A

Player B

1

1

1

2

2

2

1

2

3

4

4

2

-1

-1

Types of Games

Symmetric games

Zero Sum games

Cooperative games

Imperfect Information games

Continuous games

Symmetric Games

Strategies of both players are the sameCommon in many classical 2x2 games such as the Prisoners DilemmaNash equilibrium is where both confess and betray the otherBoth have the same strategy: Always choose to confess

1, 1 10, 0 0, 10 5, 5

Prisoner B

Prisoner A

Confess

Confess

Not Confess

Not Confess

Zero Sum Games

Game in which all payoffs add to zeroExample: Matching pennies gameEach player chooses either odd or even before flipping their pennies simultaneouslyIf both pennies come up either heads or tails, Even wins. Otherwise Odd wins*Notice the total sum of the payoffs = 0

1, -1 -1, 1 -1, 1 1, -1

Heads

Tails

Heads

Tails

Even

Odd

Cooperative Games

A game is cooperative if the players are able to form binding commitmentsCommunication among players is allowed in cooperative gamesPlayers coordinate their strategies to attain the maximum combined payoff

3, 3 0, 5 5, 0 1, 1

Player B

Player A

Defect

Defect

Cooperate

Cooperate

Imperfect information

Using earlier example, except now Player B does not know Player A’s choice of actionIn this case Player B will be tempted to choose option 2 to get a payoff of 4 (assuming Player A chooses option 1), not knowing A’s strategy

1

1

1

2

2

2

1

2

3

4

4

0

-1

-1

Player A

Player B

Continuous Games

Games in which there is not a discrete number of players, moves, and/or outcomes

The strategy set for each player is also continuous

Example: Cops and Robbers (pursuit & evasion game)

A group of players trying to capture another group (the number of players varies)

Game does not have a finite length or outcome (some robbers may never get caught)

Applications

Economics

Bargaining, duopolies, fair division, etc.

Political Science

Political economy, public choice, social choice theory, etc.

Biology

Animal behavior

Computer Science & Logic

Interactive computations, multi-agent systems

Philosophy

Social norms

Limitations

Assumptions made by game theorists are sometimes violated

Human behavior often deviates from game theory models due to irrationality and different motives (altruism)

What is the equilibrium outcome of this game?

Chip (C) and Dale (D) are negotiating over how to divide a pile of 100 acorns. The order of events is:

First Round: C makes D an initial offer. D accepts or rejects. If D accepts, the game ends and C and D get their acorns. If D rejects, 10 acorns rot because of the delay and the game continues with 90 acorns to be divided.

Second Round: D makes an offer. C accepts or rejects. If C accepts, the game ends and C and D get their acorns. If C rejects, 10 acorns rot because of the delay and the game continues with 80 acorns to be divided.

Third Round: C makes a final offer. D accepts or rejects. If D accepts, then C and D get their acorns. If D rejects, the game ends and neither C nor D get any acorns.

Works Cited

http://www.answers.com/topic/game-theory

http://en.wikipedia.org/wiki/Game_theory

http://plato.stanford.edu/entries/game-theory

http://william-king.www.drexel.edu/top/eco/game/zerosum.html