Game Theory

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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. ID: 200825

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Game Theory

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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
0
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
1
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
2
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)
3
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
4
Limitations
Assumptions made by game theorists are sometimes violated
Human behavior often deviates from game theory models due to irrationality and different motives (altruism)
5
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.
6
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