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

Equilibria and Complexity: What now?

Christos H. Papadimitriou

UC Berkeley

“christos”

Warwick, March 26 2007 2

Outline

• Equilibria and complexity: what, who and why

• Approximate Nash

• Special cases

• New equilibria concepts

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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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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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• 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

Why Complexity?

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Equilibria: the trade-offs

efficiency

existence

naturalness

correlated

pure Nash

mixed Nash[DGP06, CD06]

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Equilibria: the succinct case

efficiency

existence

naturalness

correlated[PR SODA-STOC05]

pure NashNP-c/PLS-c [FPT03]

mixed Nash[DFP ICALP06]

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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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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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Exponential directed graphwith indegree, outdegree < 2

Standard source(given)

?(there mustbe a sink…)

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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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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

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“multiplication is the name of the game and each generation plays the same”

Bobby Darren, 1961

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The multiplication game

x

y

z = x · y

“affects”

w

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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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Brouwer Nash

So….

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game over?

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What next?

efficiency

existence

naturalness

?

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-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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-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

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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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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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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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Pure equilibria

Theorem: In any anonymous game there is a pure 2s-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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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)

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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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Other equilibrium concepts:Nash dynamics

pure strategy profiles

best response(or improving response)

by one player

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“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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Unit recall equilibria

a

b

1 2

a b

1

1

22

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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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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PS: Nash dynamics and BGP oscillations

1

0

2 3

120 > 10230 > 20310 > 30

oscillation!

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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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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?