Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos”

geneva, feb ,-1-1,1 1,-1 0,00,00,10,1 1,01,0-1,-1 3,33,30,40,4 4,04,01,11,1 matching penniesprisoner’s dilemmachicken Games help us understand rational behavior in competitive situations

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

geneva, feb can it be found in polynomial time?

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

geneva, feb …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 Brouwer  Nash

geneva, feb 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 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 Correlated equilibrium 4,44,41,51,5 5,15,10,00,0 Chicken: Two pure equilibria {me, you} Mixed (½, ½) (½, ½) payoff 5/2

geneva, feb Idea (Aumann 1974) “Traffic signal” with payoff 3 Compare with Nash equilibrium Even better with payoff 3 1/3 0½ ½0 1/4 1/3 0 Probabilities in a lottery drawn by an impartial outsider, and announced to each player separately

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

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

geneva, feb 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 ns n numbers

geneva, feb 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 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, ns d numbers suffice to describe the game Also: multimatrix, congestion, location, anonymous, hypergraphical, …

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

geneva, feb 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 These k inequalities are themselves infeasible!

geneva, feb infeasible also infeasible UX T just need to solve

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

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

geneva, feb From d-graphical games to d 2 -normal-form games 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 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 “Multiplication is the name of the game and each generation plays the same…”

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

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

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

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

geneva, feb Brouwer  Nash So….

geneva, feb 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 In November… Conjecture 1: 3-player Nash is also PPAD-complete Proved!! [Chen&Deng05, DP05]

geneva, feb In December… Conjecture 2: 2-player Nash is in P PPAD-complete [Chen&Deng05b]

geneva, feb game over!