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Equilibria and Complexity: What now?

Equilibria and Complexity: What now?. Christos H. Papadimitriou UC Berkeley “christos”. Outline. Equilibria and complexity: what, who and why Approximate Nash Special cases New equilibria concepts. The basic question.

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Equilibria and Complexity: What now?

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

  2. Outline • Equilibria and complexity: what, who and why • Approximate Nash • Special cases • New equilibria concepts Warwick, March 26 2007

  3. 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.) Warwick, March 26 2007

  4. 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 Warwick, March 26 2007

  5. Why Complexity? • Equilibria are notions of rationality, aspiring models of behavior • Efficient computability is an important modeling prerequisite • “If your laptop can’t find it, neither can the market” • Furthermore: Equilibria problems raise some of the most intriguing questions in the theory of algorithms and complexity Warwick, March 26 2007

  6. Equilibria: the trade-offs mixed Nash [DGP06, CD06] correlated existence efficiency naturalness pure Nash Warwick, March 26 2007

  7. Equilibria: the succinct case mixed Nash [DFP ICALP06] correlated [PR SODA-STOC05] existence efficiency naturalness pure Nash NP-c/PLS-c [FPT03] Warwick, March 26 2007

  8. Complexity of Mixed Nash • PPAD-complete [GP, DGP] STOC 06 • Even for 3 players [CD05, DP05] • Even for 2 players (!?!) [CD] FOCS 06 Warwick, March 26 2007

  9. 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 Warwick, March 26 2007

  10. Exponential directed graphwith indegree, outdegree < 2 Standard source (given) ? (there must be a sink…) Warwick, March 26 2007

  11. 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 Warwick, March 26 2007

  12. What “PPAD-complete” mean, really? • Nash’s 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 Warwick, March 26 2007

  13. “multiplication is the name of the game and each generation plays the same” Bobby Darren, 1961 Warwick, March 26 2007

  14. The multiplication game x “affects” z = x · y w y Warwick, March 26 2007

  15. 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 Warwick, March 26 2007

  16. So…. Brouwer  Nash Warwick, March 26 2007

  17. game over? Warwick, March 26 2007

  18. What next? ? existence efficiency naturalness Warwick, March 26 2007

  19. -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]) Warwick, March 26 2007

  20. -approximate Nash: what’s known • Can be found in time nlog 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 Warwick, March 26 2007

  21. 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 Warwick, March 26 2007

  22. 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]. Warwick, March 26 2007

  23. 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” Warwick, March 26 2007

  24. 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. Warwick, March 26 2007

  25. Also: PTAS! Binomial variables x1, x2, …xn with probabilities p1, p2,…,pn They induce a distribution q = [q0, q1, …, qn] where qj = prob[∑xi =j] Lemma: There is a way to round the pi’s to multiples of 1/k so that |q - q| < O(k-1/4) Warwick, March 26 2007

  26. 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 Warwick, March 26 2007

  27. Other equilibrium concepts:Nash dynamics pure strategy profiles best response (or improving response) by one player Warwick, March 26 2007

  28. “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] Warwick, March 26 2007

  29. Unit recall equilibria 1 2 A strategy for the row player a 1 2 b 2 a b 1 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]) Warwick, March 26 2007

  30. 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?) Warwick, March 26 2007

  31. PS: Nash dynamics and BGP oscillations 1 120 > 10 230 > 20 310 > 30  oscillation! 0 2 3 Warwick, March 26 2007

  32. 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] Warwick, March 26 2007

  33. 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? Warwick, March 26 2007

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