Games Computers (and Computer Scientists) Play Avi Wigderson

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

Plan Complexity of Games Implementation of Games Design of Games Games against Clairvoyance

Complexity of Games

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

Rectangle Game m n m=4 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? 4 5 1

Zero-Sum Games Matching Pennies (simultaneous play) H HT T Strategic Form “ Best ” strategy for each player is to flip a fair coin. Game value is m n v ij -v ij i j 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.

Nash Equilibrium Chicken [Aumann] C CD D Strategic Form Probabilistic strategies (S w, S b ). Nash Equilibrium: No player has an incentive to change its strategy given the opponent’s strategy. here S w =S b = [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).

Implementing Games

The Millionaires’ Problem AliceBob BA Both want to know who is richer Neither gets any other information Question: Is that possible?

Joint random decisions CD C D Nash eq. With Independent Strategies Nash eq. With Correlated Strategies [Aumann] 3/4 1/4 3/41/4 Expected value = 3/4 Prob[CC] = 9/16 Prob[CD] = 3/16 Prob[DC] = 3/16 Prob[DD] = 1/16 Prob[CD] = 1/2 Prob[DC] = 1/2 Prob[CC] = 0 Prob[DD] = 0 Expected value = 1 Question: How to flip a coin jointly?

Simultaneity HT H T 1/2 Expected value = 0 (if they play simultaneously) Question: How do we guarantee simultaneity? xWxW xBxB A computational representation: outcome Parity Function x W x B Parity(x W, x B ) P

Privacy vs. Resilience Q 1 : How to guarantee x 1  5? Q 2 : How to guarantee x 1 remains private? Majority Function x1x1 x3x3 x 1 x 2 x 3 Majority(x 1, x 2, x 3 ) Voting M x2x2 Millionaire ’ s Problem Poker Any game

Completeness Theorem 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 [Yao, Goldreich – Micali – Wigderson]: 1.More than 1/2 of the players are honest 2.Players computationally bounded 3.Trap-door functions exist (e.g. factoring integers is hard) Hard problems can be useful!

Correct & Private digital implementation Secrets Preferences Strategies Trusted party Ideal implementation 12n s1s1 s2s2 snsn Internet Digital implementation

How to ensure Privacy Oblivious Computation [Yao] f(inputs) PMP MP P 1

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.

Where is Waldo? [Naor]

Designing Games

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 P parity M majority M MM M 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.

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]: 2 nd 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

Coping with Uncertainty Competing against Clairvoyance

On-line Problems Investor ’ s Problem (One-way trading) day price Profit/loss Muggle ’ s action Wizard ’ s action

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

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]

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

Completeness Theorems 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 [Yao, Goldreich – Micali – Wigderson]: 1.More than 1/2 are honest 2.Players computationally bounded 3.Trap-door functions exist (e.g. factoring integers is hard) Theorem [Ben-Or – Goldwasser – Wigderson]: 1 ’. 2 ’. At least 3 players, more than 2/3 are honest 3 ’. Private pairwise communication