Games Computers ( and Computer Scientists ) Play

1. Games Computers(and Computer Scientists)Play Avi Wigderson

2. Computer Science Game Theory = Information Processing by Computers Agents Games • Competing • Cooperating • Faulty • Colluding • Secretive • Adversarial Computationally Bounded Communicating Digitally

3. Plan • Complexity of Games • Implementation of Games • Design of Games • Games against Clairvoyance

4. Complexity of Games

5. Extensive Form Theorem [Zermelo]: In every finite win/lose perfect information 2-player game, White or Black can force a win. Question: Can a winning strategy be efficiently computed?

6. 1 5 3 2 4 Rectangle Game 1 m=4 n=5 m n Theorem: White has a winning strategy. Proof: Assume Black has a winning strategy. Then White can mimic it and win. Contradiction! Question: What is the winning strategy?

7. 1 -1 1 -1 -1 1 -1 1 m 1 2 j 1 2 i vij-vij n Zero-Sum Games Matching Pennies (simultaneous play) H T Strategic Form H T “Best” strategy for each player is to flip a fair coin. Game value is 0. Theorem [von Neumann ‘28]: Every 0-sum game has a (Min-Max) value. Question: Can the value, strategies be computed? Theorem [Khachian ‘80]: Yes – Efficient linear programming algorithm.

8. -3 -3 1 1 2 0 0 2 Nash Equilibrium Chicken [Aumann] C D Strategic Form C Probabilistic strategies (Sw, Sb). D Nash Equilibrium: No player has an incentive to change its strategy given the opponent’s strategy. here Sw=Sb = [C with prob ¾, D with prob ¼] Theorem [Nash]: Every (matrix) game has an equilibrium. Question: Can the players compute (any) equilibrium? Best known algorithm: exponential time (infeasible).

9. Implementing Games

10. A B Alice Bob The Millionaires’ Problem Both want to know who is richer Neither gets any other information Question: Is that possible?

11. -3 -3 1 1 2 0 0 2 C D 3/4 1/4 3/4 C Expected value = 3/4 Prob[CC] = 9/16 Prob[CD] = 3/16 Prob[DC] = 3/16 Prob[DD] = 1/16 1/4 D Prob[CD] = 1/2 Prob[DC] = 1/2 Prob[CC] = 0 Prob[DD] = 0 Expected value = 1 Joint random decisions Nash eq. With Independent Strategies Nash eq. With Correlated Strategies [Aumann] Question: How to flip a coin jointly?

12. 1/2 H 1/2 T Expected value = 0 (if they play simultaneously) 1 -1 1 -1 -1 1 -1 1 1/2 H 1/2 T A computational representation: outcome xW xB Parity(xW, xB ) 00 0 1 1 0 01 1 10 1 Parity Function P xW xB Simultaneity Question: How do we guarantee simultaneity?

13. Privacy vs. Resilience x1 x2 x3 Majority(x1, x2, x3) 00 0 0 0 01 0 0 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 Majority Function • Voting M x1 x2 x3 Q1: How to guarantee x15? Q2: How to guarantee x1 remains private? • Millionaire’s Problem • Poker • Any game

14. Completeness Theorem • Theorem [Yao, Goldreich –Micali –Wigderson]: • More than 1/2 of the players are honest • Players computationally bounded • Trap-door functions exist (e.g. factoring integers is hard) Every game, with any secrecy requirements, can be digitally implemented s.t. no collusion of the bad players can affect: *correctness (rules, outcome) * privacy (no information leaks) Hard problems can be useful!

15. Trusted party Ideal implementation Secrets Preferences Strategies s1 s2 sn 1 2 n Internet Digital implementation Internet Correct & Private digital implementation

16. 1 M M P P P P 1 0 0 1 0 1 0 How to ensure Privacy Oblivious Computation [Yao] f(inputs) 1 1 0 1 1 0

17. How to ensure Correctness • Definition [Goldwasser-Micali-Rackoff]: • zero-knowledge proofs: • Convincing • Reveal no information Theorem [Goldreich-Micali-Wigderson]: Every provable mathematical statement has a zero-knowledge proof. Corollary: Players can be forced to act legally, without fear of compromising secrets.

18. Where is Waldo? [Naor]

19. Designing Games

20. parity M P M M M M majority How to minimze players’influence Public Information Model [Ben-Or—Linial]: Joint random coin flipping Every good player flips, then combine Function Influence Parity 1 Majority 1/7 Iterated Majority 1/8 Theorem [Kahn—Kalai—Linial]: For every function, some player has non-proportional influence. Theorem [Alon—Naor]: There are “multi-round” functions for which no player has non-proportional influence.

21. How to achieve cooperation, efficiency, truthfulness Players (agents) are selfish • Auction • Question: How to get players to bid their true values? • Theorem [Clarke—Groves—Vickery]: • 2nd price auction achieves truthfulness. • Internet Games • Question: How to get players to cooperate? • [Nisan]: Distributed algorithmic mechanism design. • [Papadimitriou]: Algorithms, Games & the Internet New CS Issues: Pricing, incentives New GT Issues: Complexity, Algorithms

22. Coping with UncertaintyCompeting against Clairvoyance

23. price day Wizard’s action • On-line Problems • Investor’s Problem (One-way trading) 1 2 3 4 5 6 7 8 9 Profit/loss Muggle’s action

24. On-line problems are everywhere: • Computer operating systems • Taxi dispatchers • Investors’ decisions • Battle decisions

25. Competitive Analysis [Tarjan—Slator]: For every sequence of events, Bound the competitive ratio: muggle-cost(sequence) wizard-cost(sequence) Can be achieved in many settings. Huge, successful theory. “Online Computation and Competitive Analysis” [Borodin—El-Yaniv]

26. Nature Alice ... ... ... ... Bob Nature • Information Sets • Player’s action depends • only on its information set Alice ... Every Game? Any secrecy requirements? Incomplete information Game in Extensive form

27. Completeness Theorems • Theorem [Yao, Goldreich –Micali –Wigderson]: • More than 1/2 are honest • Players computationally bounded • Trap-door functions exist (e.g. factoring integers is hard) Every game, with any secrecy requirements, can be digitally implemented s.t. no collusion of the bad players can affect: *correctness (rules, outcome) * privacy (no information leaks) Theorem [Ben-Or –Goldwasser –Wigderson]: 1’. 2’. At least 3 players, more than 2/3 are honest 3’. Private pairwise communication