1. Game Theory M.Pajhouh Niya M.Ghotbi
Quantitative Analysis for Decision Making
M.Ghotbi
M.Pajhouh Niya
Abbas Keramati ( Assistant Professor)
University of Tehran – MBA
Fall 2008

2. 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
Title
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3. What is Game Theory? DGDG Title
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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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4. History of Game Theory DGDG Title
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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5. Applications of Game Theory
Title
Mathematics
Computer Science
Biology
Economics
Political Science
International Relations
Philosophy
Psychology
Law
Military Strategy
Management
Sports
Game Playing
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6. DGDG Key Elements of a game Title 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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7. Types of games DGDG Title
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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8. Pure Strategies DGDG Title
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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9. An Example:
Title A B Y1 Y2
Minimum X1 10 6 X2
-12 7
Maximum
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10. Mixed Strategies DGDG Title
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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11. DGDG Nash Equilibrium (NE) Title
A player’s best strategy is that strategy that maximizes that player’s payoff (utility), knowing the strategy's of the other players.
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12. DGDG 2-players Zero-Sum games Title 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.
Title
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Player 2
Tail
Head
-1 , 1
1 , -1
Player 1

13. DGDG Prisoner’s Dilemma Title
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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14. Payoff Matrix for Prisoner’s Dilemma
Title
Ted
Confess Not Confess
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Both get 5 years
1 year for Bill
10 years for Ted
10 years for Bill
1 year for Ted
Both get 3
years
Confess
Bill
Not Confess
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15. DGDG An Example of Mixed Strategy game TitleB
B believes
B doesn’t believe
A bluffs 1
A doesn’t bluff
0.5
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16. DGDG Now let’s Play This Game Title Pirate GameC B
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17. Nash’s Equilibrium DGDG Title
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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18. Definition: Nash Equilibrium
Title
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Given others’ choices, player i cannot be better-off if she deviates from si*
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19. 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
Title
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20. DGDG Title Refrences Dixit and Nalebuff: Thinking Strategically Dutta:
Strategies and Games: Theory and Practice
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