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Introduction to Game Theory Yale Braunstein Spring 2007
General approach ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Brief History of Game Theory ,[object Object],[object Object],[object Object],[object Object]
Rationality ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Utility Theory ,[object Object],[object Object],[object Object],[object Object],[object Object]
What is Game Theory? ,[object Object]
Game Theory ,[object Object],[object Object],[object Object],[object Object]
Why is game theory important? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Types of Games ,[object Object],[object Object],[object Object],[object Object],[object Object]
Zero-Sum Games ,[object Object],[object Object],[object Object]
Non-zero Sum Game ,[object Object],[object Object],[object Object]
Games of Perfect Information ,[object Object],[object Object]
Imperfect Information ,[object Object],[object Object]
Key Area of Interest ,[object Object],[object Object],Non-zero Sum Imperfect Information
Matrix Notation Notes: Player I's strategy A may be different from Player II's. P2 can be omitted if zero-sum game
Prisoner’s Dilemma & Other famous games A sample of other games: Marriage Disarmament (my generals are more irrational than yours)
Prisoner’s Dilemma 10 , 10 Blame Don't Blame Don't 20 , 0 0 , 20 1 , 1 Prisoner 1 Prisoner 2 Notes: Higher payoffs (longer sentences) are bad. Non-cooperative equilibrium  Joint maximum Institutionalized “solutions” (a la criminal organizations, KSM) NCE Jt. max.
Games of Conflict ,[object Object],[object Object],[object Object]
Games of Co-operation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Prisoner’s Dilemma with Iteration ,[object Object],[object Object],[object Object],[object Object]
Basic Strategies ,[object Object],[object Object],[object Object],[object Object],[object Object]
Plan ahead and look back Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent
If you have a dominating strategy, use it Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Use strategy 1
Eliminate any dominated strategy Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Eliminate strategy 2 as it’s dominated by strategy 1
Look for any equilibrium ,[object Object],[object Object],[object Object]
Maximin & Minimax Equilibrium ,[object Object],[object Object],[object Object]
Maximin & Minimax Equilibrium Strategies Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Min 1000 150 - 10 -15 160 Max
Definition: Nash Equilibrium ,[object Object],[object Object]
Is this a Nash Equilibrium? Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Min 1000 150 - 10 -15 160 Max
Cost to press button = 2 units When button is pressed, food given = 10 units Boxed Pigs Example
5 , 1 Press Wait Press Wait 9 , -1 4 , 4 0 , 0 Little Pig Big Pig Decisions, decisions...
Time for "real-life" decision making ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Mixed Strategy Safe 2 Safe 1 $ 0 $10,000 $100,000 Safe 1 Safe 2 $ 0
Mixed Strategy Solution
The Payoff Matrix for Holmes & Moriarity Player #1 Player #2 Strategy #1 Strategy #2 Strategy #1 Strategy #2 Payoff (1,1) Payoff (1,2) Payoff (2,1) Payoff (2,2) Canterbury Canterbury Dover Dover 0 100 50 0 Holmes Moriarty
Evolutionary Game Theory ,[object Object],[object Object],[object Object]
Hawk / Dove Game
Evolutionary Stable Strategy ,[object Object],[object Object],[object Object]
Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
 
Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
 
Where is game theory currently used? ,[object Object],[object Object],[object Object]
Limitations & Problems ,[object Object],[object Object],[object Object],[object Object]
Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Sources ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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GT-Intro to Game Theory

  • 1. Introduction to Game Theory Yale Braunstein Spring 2007
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  • 15. Matrix Notation Notes: Player I's strategy A may be different from Player II's. P2 can be omitted if zero-sum game
  • 16. Prisoner’s Dilemma & Other famous games A sample of other games: Marriage Disarmament (my generals are more irrational than yours)
  • 17. Prisoner’s Dilemma 10 , 10 Blame Don't Blame Don't 20 , 0 0 , 20 1 , 1 Prisoner 1 Prisoner 2 Notes: Higher payoffs (longer sentences) are bad. Non-cooperative equilibrium  Joint maximum Institutionalized “solutions” (a la criminal organizations, KSM) NCE Jt. max.
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  • 22. Plan ahead and look back Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent
  • 23. If you have a dominating strategy, use it Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Use strategy 1
  • 24. Eliminate any dominated strategy Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Eliminate strategy 2 as it’s dominated by strategy 1
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  • 27. Maximin & Minimax Equilibrium Strategies Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Min 1000 150 - 10 -15 160 Max
  • 28.
  • 29. Is this a Nash Equilibrium? Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160 Min 1000 150 - 10 -15 160 Max
  • 30. Cost to press button = 2 units When button is pressed, food given = 10 units Boxed Pigs Example
  • 31. 5 , 1 Press Wait Press Wait 9 , -1 4 , 4 0 , 0 Little Pig Big Pig Decisions, decisions...
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  • 33. Mixed Strategy Safe 2 Safe 1 $ 0 $10,000 $100,000 Safe 1 Safe 2 $ 0
  • 35. The Payoff Matrix for Holmes & Moriarity Player #1 Player #2 Strategy #1 Strategy #2 Strategy #1 Strategy #2 Payoff (1,1) Payoff (1,2) Payoff (2,1) Payoff (2,2) Canterbury Canterbury Dover Dover 0 100 50 0 Holmes Moriarty
  • 36.
  • 37. Hawk / Dove Game
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  • 39. Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
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  • 41. Hawk Dove Hawk Dove 2 2 10 0 10 0 -5 -5
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