game theory presentation
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Abbas Keramati ( Assistant Professor) University of Tehran – MBA Fall 2008
Game Theory Quantitative Analysis for Decision Making
M.Ghotbi
M.Pajhouh Niya
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Outline• What is Game Theory?• History of Game Theory• Applications of Game Theory• Key Elements of a game• Types of games• Nash Equilibrium (NE)• Pure Strategies & Mixed Strategies• 2players Zero-Sum games
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What is Game Theory?What is Game Theory?
• In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.
• The mathematics of human interactions
• In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose.
• The mathematics of human interactions
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History of Game Theory• von Neumann wrote a key paper in 1928• 1944: “Theory of Games and Economic
Behavior” by von Neumann and Morgenstern• 1950: Nash invents concept of Nash
equilibrium• Game theory booms after this…• 1994: Harsanyi, Nash, and Selten win Nobel
Prize in economics for game theory work
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• Mathematics• Computer Science• Biology• Economics• Political Science• International Relations• Philosophy
• Psychology• Law• Military Strategy• Management• Sports• Game Playing
Applications of Game Theory
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Key Elements of a game
• Players: Who is interacting?
• Strategies: What are their options?
• Payoffs: What are their incentives?
• Information: What do they know?
• Rationality: How do they think?
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Types of games
• Cooperative or non-cooperative
• Zero sum and non-zero sum
• Simultaneous and sequential
• Perfect information and imperfect information
• Finite & Infinite Strategies
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Pure Strategies
•The upper value of the game is equal to the minimum of the maximum values in the columns.
•The lower value of the game is equal to the maximum of the minimum values in the rows.
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A
BY1 Y2 Minimum
X1 10 6 6
X2 -12 7 -12
Maximum 10 767
An Example:
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Mixed Strategies
A mixed strategy game exists when there is no saddle point. Each player will then optimize their expected gain by determining the percent of time to use each strategy.
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Nash Equilibrium (NE)
A player’s best strategy is A player’s best strategy is that strategy that maximizes that strategy that maximizes that player’s payoff (utility), that player’s payoff (utility), knowing the strategy's of the knowing the strategy's of the other players.other players.
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2-players Zero-Sum games
-1 , 1 1 , -1
1 , -1 -1 , 1Player 1
Player 2
Penny Matching: • Each of the two players has a penny.
• Two players must simultaneously choose whether to show the Head or the Tail.
• Both 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.
Tail
Head
Tail
Head
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Prisoner’s Dilemma
• No 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 one.
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Ted
Confess Not Confess
Confess
Bill
Not Confess
Both get 5 years
1 year for Bill 10 years for
Ted
10 years for Bill
1 year for TedBoth get 3
years
Payoff Matrix for Prisoner’s Dilemma
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An Example ofMixed Strategy game
A
BB believes B doesn’t
believe
A bluffs 1 0
A doesn’t bluff 0 0.5
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Now let’s Play This Game
Pirate Game
A
B
D
C
E
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Nash’s Equilibrium • This equilibrium occurs when each player’s strategy is
optimal, knowing the 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.
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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,
),...,,,,...,(
),...,,,,...,(
**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
Given others’ choices, player i cannot be better-off if she deviates from si*
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Nash’s Equilibrium cont.: Bayesian Nash Equilibrium
• The Nash Equilibrium of the imperfect-information game
– 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 by nature.
– All players have the same prior beliefs about the probability distribution on 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
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• Dixit and Nalebuff: Thinking Strategically • Dutta: Strategies and Games: Theory and Practice • www.gametheory.net
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