Computing Equilibria

1. Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos”

2. Games help us understand rational behavior in competitive situations matching pennies chicken prisoner’s dilemma Khachiyan lecture, March 2 06

3. 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

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

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

6. …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

7. 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

8. 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

9. An Easier Problem:Correlated equilibrium Chicken: • Two pure equilibria {me, you} • Mixed (½, ½) (½, ½)payoff 5/2 Khachiyan lecture, March 2 06

10. Idea (Aumann 1974) • “Traffic signal” with payoff 3 • Compare with Nash equilibrium • Even better with payoff 3 1/3 Probabilities in a lottery drawn by an impartial outsider, and announced to each player separately Khachiyan lecture, March 2 06

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

12. 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

13. Linear programming! • n players, s strategies each • ns2 inequalites • sn variables! • Nice for 2 or 3 players • But many players? Khachiyan lecture, March 2 06

14. 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 Khachiyan lecture, March 2 06

15. 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 • multi-player • succinctly representable Khachiyan lecture, March 2 06

16. 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, … Khachiyan lecture, March 2 06

17. 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

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

19. Lemma [Hart and Schmeidler, 89]: For every y there is an x such that xUTy = 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

20. Leonid Khachiyan [1953-2005] Khachiyan lecture, March 2 06

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

22. also infeasible infeasible UXT just need to solve Khachiyan lecture, March 2 06

23. 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

24. 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

25. 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

26. 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

27. From d-graphical games to d2-normal-form games • Color the graph with d2 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

28. 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

29. “Multiplicationis the name of the gameand each generationplays the same…” Khachiyan lecture, March 2 06

30. The multiplication game x z wins when it plays 1 and w plays 0 “affects” z = x · y w if w plays 0, then it gets xy. if it plays 1, then it gets z, but z gets punished y Khachiyan lecture, March 2 06

31. 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

32. 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

33. 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

34. 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

35. 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

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

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

38. game over! Khachiyan lecture, March 2 06

39. Thank You! Khachiyan lecture, March 2 06