Game Theory and Computational Complexity Paul W. Goldberg1 1Department of Computer Science University of Liverpool, U. K. EWSCS'09 Winter School, Estonia March 2009 Goldberg Game Theory and Computational Complexity Total search problems, PPAD and related complexity classes PPAD-completeness of finding unrestricted Nash equilibria [Daskalakis, G and Papadimitriou, CACM Feb'09] The quest for fast algorithms to compute approximate Nash equilibria and I really ought to mention \Potential games" and the complexity class PLS, but won't have time to do the details. Topics Problem statements, Nash equilibrium NP-completeness of finding certain Nash equilibria1 1I will give you definitions soon! Goldberg Game Theory and Computational Complexity The quest for fast algorithms to compute approximate Nash equilibria and I really ought to mention \Potential games" and the complexity class PLS, but won't have time to do the details. Topics Problem statements, Nash equilibrium NP-completeness of finding certain Nash equilibria1 Total search problems, PPAD and related complexity classes PPAD-completeness of finding unrestricted Nash equilibria [Daskalakis, G and Papadimitriou, CACM Feb'09] 1I will give you definitions soon! Goldberg Game Theory and Computational Complexity Topics Problem statements, Nash equilibrium NP-completeness of finding certain Nash equilibria1 Total search problems, PPAD and related complexity classes PPAD-completeness of finding unrestricted Nash equilibria [Daskalakis, G and Papadimitriou, CACM Feb'09] The quest for fast algorithms to compute approximate Nash equilibria and I really ought to mention \Potential games" and the complexity class PLS, but won't have time to do the details. 1I will give you definitions soon! Goldberg Game Theory and Computational Complexity Goldberg Game Theory and Computational Complexity Common motivations: modeling rationality (interaction of selfish agents on Internet); AI: solve cognitive tasks such as negotiation It turns out that GT gives rise to problems that pose very interesting mathematical challenges, e.g. w.r.t. computational complexity. Complexity classes PPAD and PLS. Game Theory and Computer Science Modern CS and GT originated with John von Neumann at Princeton in the 1950s [Yoav Shoham, CACM Aug'08] Goldberg Game Theory and Computational Complexity It turns out that GT gives rise to problems that pose very interesting mathematical challenges, e.g. w.r.t. computational complexity. Complexity classes PPAD and PLS. Game Theory and Computer Science Modern CS and GT originated with John von Neumann at Princeton in the 1950s [Yoav Shoham, CACM Aug'08] Common motivations: modeling rationality (interaction of selfish agents on Internet); AI: solve cognitive tasks such as negotiation Goldberg Game Theory and Computational Complexity Game Theory and Computer Science Modern CS and GT originated with John von Neumann at Princeton in the 1950s [Yoav Shoham, CACM Aug'08] Common motivations: modeling rationality (interaction of selfish agents on Internet); AI: solve cognitive tasks such as negotiation It turns out that GT gives rise to problems that pose very interesting mathematical challenges, e.g. w.r.t. computational complexity. Complexity classes PPAD and PLS. Goldberg Game Theory and Computational Complexity e.g. Oil cartel cut production raise production 0 1 0 cut production 1 raise production Solution: both players defect. Numbers in red are probabilities. Example 1: Prisoners' dilemma cooperate defect cooperate 8 10 8 0 defect 0 1 10 1 There's a row player and a column player. Nash equilibrium: no incentive to change Goldberg Game Theory and Computational Complexity e.g. Oil cartel cut production raise production cut production raise production Example 1: Prisoners' dilemma cooperate defect 0 1 cooperate 0 8 10 8 0 defect 1 0 1 10 1 There's a row player and a column player. Solution: both players defect. Numbers in red are probabilities. Nash equilibrium: no incentive to change Goldberg Game Theory and Computational Complexity cooperate defect cooperate defect Example 1: Prisoners' dilemma e.g. Oil cartel cut production raise production 0 1 0 cut production 8 10 8 0 1 raise production 0 1 10 1 There's a row player and a column player. Solution: both players defect. Numbers in red are probabilities. Nash equilibrium: no incentive to change Goldberg Game Theory and Computational Complexity Example 2: Rock-paper-scissors 2008 Rock-paper-scissors Championship (Las Vegas, USA) Goldberg Game Theory and Computational Complexity 1/3 1/3 1/3 1/3 1/3 1/3 Solution: both players randomize: probabilities are shown in red. Rock-paper-scissors: payoff matrix rock paper scissors rock 0 1 -1 0 -1 1 paper -1 0 1 1 0 -1 scissors 1 -1 0 -1 1 0 Goldberg Game Theory and Computational Complexity Rock-paper-scissors: payoff matrix rock paper scissors 1/3 1/3 1/3 rock 1/3 0 1 -1 0 -1 1 paper 1/3 -1 0 1 1 0 -1 scissors 1/3 1 -1 0 -1 1 0 Solution: both players randomize: probabilities are shown in red. Goldberg Game Theory and Computational Complexity 1/3 5/12 1/4 1/3 1/3 1/3 (thanks to Rahul Savani's on-line Nash equilibrium solver.) Rock-paper-scissors: a non-symmetrical variant rock paper scissors rock 0 1 -1 0 -1 2 paper -1 0 1 1 0 -1 scissors 1 -1 0 -1 1 0 What is the solution? Goldberg Game Theory and Computational Complexity Rock-paper-scissors: a non-symmetrical variant rock paper scissors 1/3 5/12 1/4 rock 1/3 0 1 -1 0 -1 2 paper 1/3 -1 0 1 1 0 -1 scissors 1/3 1 -1 0 -1 1 0 What is the solution? (thanks to Rahul Savani's on-line Nash equilibrium solver.) Goldberg Game Theory and Computational Complexity It takes 2 hunters to catch a stag, but only one to catch a rabbit. Example 3: Stag hunt 2 hunters; each chooses whether to hunt stag or rabbit... Goldberg Game Theory and Computational Complexity but only one to catch a rabbit. Example 3: Stag hunt 2 hunters; each chooses whether to hunt stag or rabbit... It takes 2 hunters to catch a stag, Goldberg Game Theory and Computational Complexity Example 3: Stag hunt 2 hunters; each chooses whether to hunt stag or rabbit... It takes 2 hunters to catch a stag, but only one to catch a rabbit. Goldberg Game Theory and Computational Complexity 0 1/8 1 7/8 0 1/8 1 7/8 Or, both players hunt rabbit. Or, both players randomize (with the right probabilities). Stag hunt: payoff matrix hunt stag hunt rabbit 1 0 hunt stag 1 8 1 8 0 hunt rabbit 0 0 1 1 1 Solution: both hunt stag (the best solution). Goldberg Game Theory and Computational Complexity 1 1/8 0 7/8 1 1/8 0 7/8 Or, both players randomize (with the right probabilities). Stag hunt: payoff matrix hunt stag hunt rabbit 0 1 hunt stag 0 8 1 8 0 hunt rabbit 1 0 1 1 1 Solution: both hunt stag (the best solution). Or, both players hunt rabbit. Goldberg Game Theory and Computational Complexity 1 0 0 1 1 0 0 1 Stag hunt: payoff matrix hunt stag hunt rabbit 1/8 7/8 hunt stag 1/8 8 1 8 0 hunt rabbit 7/8 0 1 1 1 Solution: both hunt stag (the best solution). Or, both players hunt rabbit. Or, both players randomize (with the right probabilities). Goldberg Game Theory and Computational Complexity Number the players1 ; 2; :::; k. Let Sp denote player p's set of actions. e.g. in rock-paper-scissors, S1 = S2 = frock; paper; scissorsg. n denotes the size of the largest Sp. (So, in rock-paper-scissors, k = 2, n = 3.) If k is a constant, we seek algorithms polynomial in n. Indeed, much work studies special case k = 2, where a game's payoffs can be written down in 2 matrices. S = S1 × S2 × ::: × Sk is the set of pure strategy profiles. i.e. if s 2 S, s denotes a choice of action, for each player. p Each s 2 S gives rise to utility or payoff to each player. us will denote the payoff to player p when all players choose s. Definition and notation Game: set of players, each player has his own set of allowed actions (also known as \pure strategies"). Any combination of actions will result in a numerical payoff (or value, or utility) for each player. (A game should specify the payoffs, for every player and every combination of actions.) Goldberg Game Theory and Computational Complexity Let Sp denote player p's set of actions. e.g. in rock-paper-scissors, S1 = S2 = frock; paper; scissorsg. n denotes the size of the largest Sp. (So, in rock-paper-scissors, k = 2, n = 3.) If k is a constant, we seek algorithms polynomial in n. Indeed, much work studies special case k = 2, where a game's payoffs can be written down in 2 matrices. S = S1 × S2 × ::: × Sk is the set of pure strategy profiles. i.e. if s 2 S, s denotes a choice of action, for each player. p Each s 2 S gives rise to utility or payoff to each player. us will denote the payoff to player p when all players choose s. Definition and notation Game: set of players, each player has his own set of allowed actions (also known as \pure strategies"). Any combination of actions will result in a numerical payoff (or value, or utility) for each player. (A game should specify the payoffs, for every player and every combination of actions.) Number the players1 ; 2; :::; k. Goldberg Game Theory and Computational Complexity n denotes the size of the largest Sp.