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)

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

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)

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

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)

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

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

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

Vickrey auction  Payoff to 1 = Prob(b 1 > b 2 )[v 1 – b 2 ] If v 1 > b 2, want to max Prob(b 1 >b 2 ) ‏ so set b 1 =v 1 If v 1 b 2 ) so set b 1 =v 1  So always want to tell the truth

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

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

Parrando’s Paradox: Outcomes

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

Witsenhausen Counterexample  Proposed in 1968  Numerically solved in 2001  See 2 papers in references

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)

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

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  Books “Game Theory and Strategy” – Peter Straffin “Game Theory” – Petrosjan and Zenkevich