1. A Little Game Theory1 A LITTLE GAME THEORY Mike Bailey MSIM 852

2. A Little Game Theory2 BASICS Two or more competitors Each chooses a strategy Pay-off determined when all strategies known John Von Newmann and Oskar Morganstern, Theory of Games and Economic Behavior (1944)  seen by many as the first publication of Operations Research  Linear Programming is introduced in a chapter

3. A Little Game Theory3 TWO-PERSON ZERO-SUM GAME Most common form Two competitors, each will be rewarded Fixed reward total  What one wins, the other loses

4. A Little Game Theory4 PAY-OFF MATRIX Presented as reward for player A xyz 1804075 2703530 B’s strategy A’s strategy

5. A Little Game Theory5 MAXIMIN (MINIMAX) A chooses the strategy where he gets the best payoff if B acts optimally  Maximizes the minimum xyz 1804075 2703530

6. A Little Game Theory6 MAXIMIN (MINIMAX) A chooses the strategy where he gets the best payoff if B acts optimally  Maximizes the minimum xyz 1804075 2703530 Does not always occur “Saddlepoint” Value of the Game

7. A Little Game Theory7 DOMINANCE y dominates x for player B xyz 1804075 2703530

8. A Little Game Theory8 DOMINANCE y dominates x for player B...then 1 dominates 2 for player A xyz 1804075 2703530

9. A Little Game Theory9 DOMINANCE y dominates x for player B...then 1 dominates 2 for player A......then y dominates z for player B xyz 1804075 2703530

10. A Little Game Theory10 DOMINANCE y dominates x for player B...then 1 dominates 2 for player A......then y dominates z for player B.........done xyz 1804075 2703530 Doesn’t always happen Useful for big tables

11. A Little Game Theory11 MIXED STRATEGIES wxyz 1751056545 270605540 380903550 4951005055

12. A Little Game Theory12 MIXED STRATEGIES wxyz 1751056545 270605540 380903550 4951005055

13. A Little Game Theory13 MIXED STRATEGIES wxyz 1751056545 270605540 380903550 4951005055

14. A Little Game Theory14 MIXED STRATEGY A will choose strategy 1 with probability p  V(y) = 65p + 50(1-p)  V(z) = 45p + 55(1-p) What value of p makes A indifferent to B’s choice? yz 16545 45055

15. A Little Game Theory15 MIXED STRATEGY A will choose strategy 1 with probability p  65p + 50(1-p) = 45p + 55(1-p) p = 0.8 V = 53 B will choose y with probability q  65q + 45(1-q) = 50q + 55(1-q) q = 0.6 V = 53 yz 16545 45055

16. A Little Game Theory16 PRISONER’S DILEMMA Payoffs are jail sentences (for A, for B) in years silentbetray silent1/2, 1/210, free betrayfree, 102,2

17. A Little Game Theory17 PRISONER’S DILEMMA Pareto Optimum  No move can make a player better off without harming another Nash Equilibrium  No player can improve payoff unilaterally silentbetray silent1/2, 1/210, free betrayfree, 102,2 http://en.wikipedia.org/wiki/Prisoner's_dilemma

18. A Little Game Theory18 APPLICATIONS ASW (Hide and Seek) Arms Control Advertising Strategy Smuggling Making the All-Star Team Multiethnic Insurgency and Revolt Drug Testing (Wired, August 2006)

19. A Little Game Theory19 ITERATED PD Set a strategy involving a sequence of choices and memory of the (choice, outcome) Random termination of the game Noise in the game Specified payoff matrix The Iterated Prisoner's Dilemma Competition: Celebrating the 20 th Anniversary http://www.prisoners-dilemma.com/