1. Equilibria and Complexity: What now? Christos H. Papadimitriou UC Berkeley christos

2. Warwick, March 26 20072 Outline Equilibria and complexity: what, who and why Approximate Nash Special cases New equilibria concepts

3. Warwick, March 26 20073 The basic question Can equilibria (of various sorts: pure Nash, mixed Nash, approximate Nash, correlated, even price equilibria) be found efficiently? Explicit games vs. succinct games (graphical, strategic form, congestion, network congestion, multimatrix, facility location, etc.)

4. Warwick, March 26 20074 The succinct game argument With games we model auctions, markets, the Internet Thus we must study multi-player games But these have exponential input Hence all games of interest are multiplayer and succinct

5. Warwick, March 26 20075 Equilibria are notions of rationality, aspiring models of behavior Efficient computability is an important modeling prerequisite If your laptop cant find it, neither can the market Furthermore: Equilibria problems raise some of the most intriguing questions in the theory of algorithms and complexity Why Complexity?

6. Warwick, March 26 20076 Equilibria: the trade-offs efficiency existence naturalness correlated pure Nash mixed Nash [DGP06, CD06]

7. Warwick, March 26 20077 Equilibria: the succinct case efficiency existence naturalness correlated [PR SODA-STOC05] pure Nash NP-c/PLS-c [FPT03] mixed Nash [DFP ICALP06]

8. Warwick, March 26 20078 Complexity of Mixed Nash PPAD-complete [GP, DGP] STOC 06 Even for 3 players [CD05, DP05] Even for 2 players (!?!) [CD] FOCS 06

9. Warwick, March 26 20079 What does PPAD-complete mean? PPAD: Class of problems that always have a solution, defined in [Pa90] Contains many well-known tough nuts (Brouwer, Borsuk-Ulam, Arrow-Debreu, Nash, …) Exponential lower bounds known for some Oracle separations from P and other classes

10. Warwick, March 26 200710 Exponential directed graph with indegree, outdegree < 2 Standard source (given) ? (there must be a sink…)

11. Warwick, March 26 200711 An aside: The four existence proofs if a directed graph has an unbalanced node, then it has another PPAD if an undirected graph has an odd-degree node, then it has another PPA every dag has a sink PLS pigeonhole principle PPP

12. Warwick, March 26 200712 What PPAD-complete mean, really? Nashs 1951 proof reduces finding a Nash equilibrium to finding a Brouwer fixpoint The proof in [DGP06] is a reduction in the opposite direction We simulate arbitrary 3-dimensional Brouwer functions by a game Main trick: games that do arithmetic

13. Warwick, March 26 200713 multiplication is the name of the game and each generation plays the same Bobby Darren, 1961

14. Warwick, March 26 200714 The multiplication game x y z = x · y affects w

15. Warwick, March 26 200715 Reduction Brouwer Nash: a very rough sketch Graphical games that do multiplication, addition, comparison, Boolean operations… Simulate the circuit that computes the Brouwer function by a huge graphical game Brittle comparator problem solved by averaging Simulate the graphical game by a 4-player game: 4-color the graph

16. Warwick, March 26 200716 Brouwer Nash So….

17. Warwick, March 26 200717 game over?

18. Warwick, March 26 200718 What next? efficiency existence naturalness ?

19. Warwick, March 26 200719 -approximate Nash a mixed strategy profile such that no player has a strategy with expected payoff bigger than the current one by more than + (assume all utilities normalized to [0,1])

20. Warwick, March 26 200720 -approximate Nash: whats known Can be found in time n log n / [LMM04] No algorithm with < 1/2 is possible, unless supports of size bigger than log n are examined [FNS07] You get = ¾ by looking at all supports of size two

21. Warwick, March 26 200721 How to do = ½ [DMP06] s is any strategy of the first player t is the best response of the other player to s s is the best response of the first player to t ½-approximate mixed strategy profile: –First player plays ½ [s + s ] –Other player plays t

22. Warwick, March 26 200722 Better than 1/2?.38 [DMP07] (by using ideas from [LMM03] plus LP) PTAS? NB: It is known that FPTAS is impossible (unless PPAD = P) [CDT06].

23. Warwick, March 26 200723 Special cases? 0-1 games are hard [AKV05] Any interesting classes for which Nash is easy? Anonymous games [DP07] Each player is different, but sees all other players as identical

24. Warwick, March 26 200724 Pure equilibria Theorem: In any anonymous game there is a pure 2 s-approximate equilibrium (where s = number of strategies, = Lipschitz constant of the utility functions) and it can be found in polynomial time.

25. Warwick, March 26 200725 Also: PTAS! Binomial variables x 1, x 2, …x n with probabilities p 1, p 2,…,p n They induce a distribution q = [q 0, q 1, …, q n ] where q j = prob[x i =j] Lemma: There is a way to round the p i s to multiples of 1/k so that |q - q | < O(k -1/4 )

26. Warwick, March 26 200726 PTAS (cont.) Now, the mixed strategies with probabilities 0, 1/k, 2/k, …, 1 can be considered as k+1 pure strategies => O(n ^(-4) ) PTAS

27. Warwick, March 26 200727 Other equilibrium concepts: Nash dynamics pure strategy profiles best response (or improving response) by one player

28. Warwick, March 26 200728 Equilibrium concept Sink strongly connected component (cf [GMV 05]) Generalizes pure Nash, always exists Expected payoff (but which trans. prob.?) How hard is this to compute? Answer: In P for normal form games, PSPACE-complete for graphical games [FP07]

29. Warwick, March 26 200729 Unit recall equilibria a b 12 ab 1 1 2 2 A strategy for the row player Problem: given a game, is there a pure Nash equilibrium in the automaton game? (Unit recall equilibrium, or URE) Could it be in P? (It is in NP [FP])

30. Warwick, March 26 200730 Componentwise unit recall equilibria (CURE) Joint work in progress with Alex Fabrikant Equilibrium if players can only change one transition at a time Universal Efficiently computable (But are they natural/convincing?)

31. Warwick, March 26 200731 PS: Nash dynamics and BGP oscillations 1 0 2 3 120 > 10 230 > 20 310 > 30 oscillation!

32. Warwick, March 26 200732 BGP oscillations (continued) Well-looked at problem in Internet theory Necessary condition (NP-complete) Sufficient condition (coNP-complete) Surprise! This is actually a Nash dynamics problem… PSPACE-complete [FP07]

33. Warwick, March 26 200733 So… The complexity of Nash leads to exciting new problems …and a rethinking of the equilibrium idea PTAS for Nash? Multiplicative version? Credible/natural, guaranteed to exist and efficiently computable equilibrium concept related to Nash dynamics?