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Game Theory

Game Theory. April 9, 2009. Prisoner’s Dilemma. One-shot, simultaneous game Nash Equilibrium (individually rational strategies) is not Pareto Optimal (group rationality) Also interesting Repeated plays (finite, infinite) Computer program competition (Tit-for-tat).

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Game Theory

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  1. Game Theory April 9, 2009

  2. Prisoner’s Dilemma • One-shot, simultaneous game • Nash Equilibrium (individually rational strategies) is not Pareto Optimal (group rationality) • Also interesting • Repeated plays (finite, infinite) • Computer program competition (Tit-for-tat)

  3. Fields where game theory is used • Engineering • Economics, Business • Auctions, voting systems, oligopolies • Biology • Evolutionary forces • Animal behavior • Evolutionary game theory • Political Science • Arms races, democratic peace • Social choice theory • Psychology • Behavioral economics • Human behavior, rationality vs irrationality

  4. Game Theory: Overview • Decision makers’ objectives • Cooperative • Same objectives, coalitions • Non-cooperative • Zero-sum games – diametrically opposed players • Non-zero-sum games • Game moves • One-shot games, repeated games, infinite games • Simultaneous vs. sequential (dynamic)

  5. Games • Information • Perfect information (Ultimatum game, Chess) • All players know previous moves • Only sequential games • Imperfect information • Complete information • known strategies and payoffs, not actions • Strategies • Continuous • Discrete

  6. Game Representation • Extensive form • More general • Sequential games • Imperfect information • Trees • Normal Form • Matrix representation showing strategies and payoffs for each player • Simultaneous games (or no knowledge of others’ move)

  7. Game Types • Combinatorial games • 2 player games • Take turns to try to win • Perfect information • Impartial games (chess, etc.) • Same moves available to all players • Partisan games • Some players have more moves than others • Classical game theory

  8. Auctions • Value of item • Private Value • Common Value • Complete information (10$ bill) • Incomplete information (jar of coins) • Procedures for bidding • Open – repeated bidding, awareness of others’ bids • Closed – sealed bids

  9. Auction Types • English - ascending price, highest wins • Dutch – descending price • Sealed bid 1st price • Depends on what you think other people will do • Bid less than value • Vickrey – sealed bid, 2nd price • True value

  10. Vickrey auction • Payoff to 1 = Prob(b1 > b2)[v1 – b2] • If v1 > b2 , want to max Prob(b1>b2)‏ so set b1=v1 • If v1 < b2 , want to min Prob(b1>b2) so set b1=v1 • So always want to tell the truth

  11. Auctions • Applications • Estate sales/antiques • E-bay • Google advertisements • Offshore oil fields • Winner’s curse – overpay for common value item with incomplete information • Bid shading (bid below value) • Cheating • Collusion – bidders form a “ring” • Unofficial auction after win, split the difference in prices • Chandelier Bidding – false bids, “Off The Wall” bidding

  12. Parrondo’s “Apparent” Paradox • Two losing games, when combined, yield a winning game • See “A Review of Parrondo’s Paradox”

  13. Parrando’s Paradox: Outcomes

  14. Links to Control Theory • “Game-theoretic approaches” occur often • Path-planning (LaValle, 2000) • Witsenhausen problem ( • Cooperative Multi-agent systems • Differential Games (next time) • Dynamic chases etc. • Missile Defense • Airplane Safety

  15. Witsenhausen Counterexample • Proposed in 1968 • Numerically solved in 2001 • See 2 papers in references

  16. Witsenhausen

  17. Witsenhausen • Learning approach • A distributed algorithm designed to find Nash Equilibria in games • 2009 Paper • Formulate problem as a potential game and use a learning algorithm to find an efficient controller • Fading memory joint strategy fictitious play (JSFP) with inertia)

  18. Game Theory in the News • 2008-Freakonomics analyses the game theory behind Beauty and the Geek • Why cooperative and competitive behavior does not die out as species evolve • Game theory gives hope to global warming solutions • Newer concepts: • Quantum game theory

  19. References • “A Review of Parrondo’s Paradox” G. Harmer and D. Abbott, 2002 • “Learning Approaches to the Witsenhausen Counterexample from a View of Potential Games” Li et al. In submission 2009 • “Review of the Witsenhausen Problem” Ho et al. 2008 • Books • “Game Theory and Strategy” – Peter Straffin • “Game Theory” – Petrosjan and Zenkevich

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