Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos”

Khachiyan lecture, March ,-1-1,1 1,-1 4,44,41,51,5 5,15,10,00,0 3,33,30,40,4 4,04,01,11,1 matching penniesprisoner’s dilemmachicken Games help us understand rational behavior in competitive situations

Khachiyan lecture, March 2 06 Concepts of rationality Nash equilibrium (or double best response) Problem: may not exist Idea: randomized Nash equilibrium Theorem [Nash 1951]: Always exists

Khachiyan lecture, March 2 06 can it be found in polynomial time?

Khachiyan lecture, March 2 06 is it then NP-complete? No, because a solution always exists

Khachiyan lecture, March 2 06 …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…”

Khachiyan lecture, March 2 06 Complexity of Nash Equilibria? 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]

Khachiyan lecture, March 2 06 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)

Khachiyan lecture, March 2 06 An Easier Problem: 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

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 Correlated equilibria Always exist (Nash equilibria are examples) Can be found (and optimized over) efficiently by linear programming

Khachiyan lecture, March 2 06 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':

Khachiyan lecture, March 2 06 Linear programming! n players, s strategies each ns 2 inequalites s n variables! Nice for 2 or 3 players But many players?

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 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, …

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 U need to show dual is infeasible show it is unbounded

Khachiyan lecture, March 2 06 Lemma [Hart and Schmeidler, 89]: For every y there is an x such that xU T y = 0 and in fact, x is the product of the steady-state distributions of the Markov chains implied by y Idea: run “ellipsoid against hope”

Khachiyan lecture, March 2 06 Leonid Khachiyan [ ]

Khachiyan lecture, March 2 06 These k inequalities are themselves infeasible!

Khachiyan lecture, March 2 06 infeasible also infeasible UX T just need to solve

Khachiyan lecture, March 2 06 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.

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 Corollaries: Graphical games (on any graph!) Polymatrix games Hypergraphical games Congestion games and local effect games Facility location games Anonymous games Etc…

Khachiyan lecture, March 2 06 Back to 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

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 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!

Khachiyan lecture, March 2 06 “Multiplication is the name of the game and each generation plays the same…”

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 Finally, 4 players Previous reduction creates a bipartite graph of degree 3 Carefully simulate each side by two players, refining the previous reduction

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 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

Khachiyan lecture, March 2 06 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 Markakis and Mehta 2003]

Khachiyan lecture, March 2 06 In November… Conjecture 1: 3-player Nash is also PPAD-complete Proved!! [Chen&Deng05, DP05]

Khachiyan lecture, March 2 06 In December… Conjecture 2: 2-player Nash is in P PPAD-complete [Chen&Deng05b]

Khachiyan lecture, March 2 06 game over!

Khachiyan lecture, March 2 06 Thank You!