Computing Equilibria Christos H Papadimitriou UC Berkeley christos

Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos” geneva, feb 14 2006

Games help us understand rational behavior in competitive situations matching pennies chicken prisoner’s dilemma 1, -1 -1, 1 0, 0 0, 1 3, 3 0, 4 -1, 1 1, -1 1, 0 -1, -1 4, 0 1, 1 geneva, feb 14 2006

Concepts of rationality • Nash equilibrium (or double best response) • Problem: may not exist • Idea: randomized Nash equilibrium Theorem [Nash 1951]: Always exists. . geneva, feb 14 2006

can it be found in polynomial time? geneva, feb 14 2006

is it then NP-complete? No, because a solution always exists geneva, feb 14 2006

…and why bother? (a parenthesis) • Equilibrium concepts provide some of the most intriguing specimens of problems • They are notions of rationality, aspiring models of behavior • Efficient computability is an important modeling prerequisite “if your laptop can’t find it, then neither can the market…” geneva, feb 14 2006

Brouwer geneva, feb 14 2006 Nash

Complexity? • Nash’s existence proof relies on Brouwer’s fixpoint theorem • Finding a Brouwer fixpoint is a hard problem • Not quite NP-complete, but as hard as any problem that always has an answer can be… • Technical term: PPAD-complete [P 1991] geneva, feb 14 2006

Complexity? (cont. ) • But how about Nash? • Is it as hard as Brouwer? • Or are the Brouwer functions constructed in the proof specialized enough so that fixpoints can be computed? (cf contraction maps) geneva, feb 14 2006

Correlated equilibrium Chicken: 4, 4 1, 5 5, 1 0, 0 • Two pure equilibria {me, you} • Mixed (½, ½) payoff 5/2 geneva, feb 14 2006

Idea (Aumann 1974) • “Traffic signal” with payoff 3 • Compare with Nash equilibrium • Even better with payoff 3 1/3 0 ½ ½ 0 1/4 1/4 1/3 1/3 0 geneva, feb 14 2006 Probabilities in a lottery drawn by an impartial outsider, and announced to each player separately

Correlated equilibria • Always exist (Nash equilibria are examples) • Can be found (and optimized over) efficiently by linear programming geneva, feb 14 2006

Linear programming? • A variable x(s) for each box s • Each player does not want to deviate from the signal’s recommendation – assuming that the others will play along • For every player i and any two rows of boxes s, s': geneva, feb 14 2006

Linear programming! • • • n players, s strategies each ns 2 inequalites sn variables! Nice for 2 or 3 players But many players? geneva, feb 14 2006

The embarrassing subject of many players • • • With games we are supposed to model markets and the Internet These have many players To describe a game with n players and s strategies per player you need nsn numbers geneva, feb 14 2006

The embarrassing subject of many players (cont. ) • These important games cannot require astronomically long descriptions “if your problem is important, then its input cannot be astronomically long…” • Conclusion: Many interesting games are 1. multi-player 2. succinctly representable geneva, feb 14 2006

e. g. , Graphical Games • [Kearns et al. 2002] Players are vertices of a graph, each player is affected only by his/her neighbors • If degrees are bounded by d, nsd numbers suffice to describe the game • Also: multimatrix, congestion, location, anonymous, hypergraphical, … geneva, feb 14 2006

Surprise! Theorem: A correlated equilibrium in a succinct game can be found in polynomial time provided the utility expectation over mixed strategies can be computed in polynomial time. Corollaries: All succinct games in the literature geneva, feb 14 2006

U show it is unbounded need to show dual is infeasible geneva, feb 14 2006

Lemma [Hart and Schmeidler, 89]: • and in fact, x is the product of the steady-state distributions of the Markov chains implied by y • hence: run “ellipsoid against hope” geneva, feb 14 2006

These k inequalities are themselves infeasible! geneva, feb 14 2006

infeasible also infeasible UXT just need to solve geneva, feb 14 2006

as long as we can solve… given a succinct representation of a game, and a product distribution x, find the expected utility of a player, in polynomial time. geneva, feb 14 2006

And it so happens that… …in all known cases, this problem can be solved by applying one, two, or all three of the following tricks: • Explicit enumeration • Dynamic programming • Linearity of expectation geneva, feb 14 2006

Corollaries: • • Graphical games (on any graph!) Polymatrix games Hypergraphical games Congestion games and local effect games Facility location games Anonymous games Etc… geneva, feb 14 2006

Nash complexity, summary 2 -Nash 3 -Nash 4 -Nash … k-Nash … ||| 1 -Gr. Nash 2 -Gr. Nash 3 -Gr. Nash … d-Gr. Nash … Theorem (with Paul Goldberg, 2005): All these problems are equivalent geneva, feb 14 2006

From d-graphical games to -normal-form games d 2 • Color the graph with d 2 colors • No two vertices affecting the same vertex have the same color • Each color class is represented by a single player who randomizes among vertices, strategies • So that vertices are not “neglected: ” generalized matching pennies geneva, feb 14 2006

From k-normal-form games to graphical games • Idea: construct special, very expressive graphical games • Our vertices will have 2 strategies each • Mixed strategy = a number in [0, 1] (= probability vertex plays strategy 1) • Basic trick: Games that do arithmetic! geneva, feb 14 2006

“Multiplication is the name of the game and each generation plays the same…” geneva, feb 14 2006

The multiplication game x z wins when it plays 1 and w plays 0 “affects” z=x·y w y 0 0 0 1 geneva, feb 14 2006 if w plays 0, then it gets x y. if it plays 1, then it gets z, but z gets punished

From k-normal-form games to 3 -graphical games (cont. ) • At any Nash equilibrium, z = x y • Similarly for +, -, “brittle comparison” • Construct graphical game that checks the equilibrium conditions of the normal form game • Nash equilibria in the two games coincide geneva, feb 14 2006

Finally, 4 players • Previous reduction creates a bipartite graph of degree 3 • Carefully simulate each side by two players, refining the previous reduction geneva, feb 14 2006

Nash complexity, summary 2 -Nash 3 -Nash 4 -Nash … k-Nash … ||| 1 -Gr. Nash 2 -Gr. Nash 3 -Gr. Nash … d-Gr. Nash … Theorem (with Paul Goldberg, 2005): All these problems are equivalent Theorem (with Costas Daskalakis and Paul Goldberg, 2005): …and PPAD-complete geneva, feb 14 2006

Nash is PPAD-complete • Proof idea: Start from a PPAD-complete stylized version of Brouwer on the 3 D cube • Use arithmetic games to compute Brouwer functions • Brittle comparator problem solved by averaging geneva, feb 14 2006

Recall: • Nash’s theorem reduces Nash to Brouwer • This is a reduction in the opposite direction geneva, feb 14 2006

So…. Brouwer geneva, feb 14 2006 Nash

Open problems • Conjecture 1: 3 -player Nash is also PPAD-complete • Conjecture 2: 2 -player Nash can be found in polynomial time • Approximate equilibria? [cf. Lipton and Markakis 2003] geneva, feb 14 2006