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

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

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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.)

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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

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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?

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

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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]

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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

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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

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

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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

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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

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

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

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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

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Warwick, March 26 200716 Brouwer Nash So….

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Warwick, March 26 200717 game over?

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

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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])

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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

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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

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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].

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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

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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.

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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 )

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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

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

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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]

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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])

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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?)

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

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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]

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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?

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