An Introduction to Game Theory

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An Introduction toGame Theory

• Kelly Bielak
• 12/8/05

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Growing Up with Game Theory

• Rock, Paper, Scissors
• Tic-Tac-Toe
• Survivor
• Weakest Link

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What is Game Theory?

• A multi-agent decision problem
• There are numerous people contending for limited
  rewards
• They have to make certain moves on which their
  payoff depends
• Have to follow certain rules
• Each player is supposed to behave rationally-
  trying to maximize profit irrespective to what
  other players are doing
• Economics, sociology, political science, etc.

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Two Branches

• Non Co-operative
• Players work independently without assuming
  anything about what other players are doing
• Co-operative
• Players co-operate with one another
• Extensive Form Players make their moves in
  turns, there is more than one move in a game, and
  each move is known to all players
• Normal Form One shot, players make simultaneous
  moves

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But What Exactly is a Game?

• Set of players
• Set of rules
• Set of strategies
• Moves players use during game
• A rational player would chose strategies to
  maximize payoff
• Set of outcomes
• Payoff

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Example Coin Matching Game

• Set of Players- P1 P2
• Set of Rules- Each player chooses H or T
  independently. P1 wins if both selections are the
  same, otherwise P2 wins
• Set Strategies- Each can chose either H or T
• Set of Outcomes- (H,H),(T,T)P1 wins
  (H,T),(T,H)P2 wins
• Payoff- P1 wins or P2 winscan have set amounts
  for payoff
• Your Turn!

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Other Types of Games

• Zero Sum Game
• Tic-Tac-Toe
• Winner 1, Loser -1, Tie 0
• Constant Sum Game
• Chess
• Winner 1, Loser 0, Tie ½ each
• Non Zero Sum Game
• Prisoners Dilemma

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Prisoners Dilemma

• 2 players Prisoner 1 Prisoner 2
• Each has 2 possible actions Confess or Dont
  Confess
• Players chose actions simultaneously without
  knowing the action of the other
• Payoff in prison years

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Prisoners Dilemma

• If both prisoners confess, each will be given a 5
  year sentence
• If both prisoners Do Not confess, the police will
  be able to pin a part of the crime on both each
  will be given a 2 year sentence
• If one confesses and the other does not, the
  confessor will make a deal with the police go
  free while the other goes to jail for 10 years

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Your Turn!

• Think of it this way
• The goal is to figure out the other persons
  strategy use it to minimize your jail time (not
  screw the other person over)

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Strategy Behind the Dilemma(For Prisoner 1)

• If Prisoner 2 confesses
• Confessing gives P1 5 years
• Denying will give P1 10 years
• Better for P1 to confess
• If Prisoner 2 doesnt confess
• Confessing frees P1
• Denying will give P1 2 years
• Again, its better for P1 to confess

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For Those of You in Probability

• P1 confesses with probability q
• P1 assumes P2 would confess with
• probability p
• Therefore
• 5pq 0xq(1-p) 10x(1-q)p 2(1-q)(1-p) qp
  q(4p1) years

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Works Consulted

• Introduction to Game Theory
• Game Theory
• What is Game Theory?
• Games, Dilemmas, and Traps
• Prisoners' Dilemma Interactive Game