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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
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
• 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?
Warwick, March 26 2007 6
Equilibria: the trade-offs
efficiency
existence
naturalness
correlated
pure Nash
mixed Nash[DGP06, CD06]
Warwick, March 26 2007 7
Equilibria: the succinct case
efficiency
existence
naturalness
correlated[PR SODA-STOC05]
pure NashNP-c/PLS-c [FPT03]
mixed Nash[DFP ICALP06]
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 mustbe 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
y
z = x · y
“affects”
w
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
Brouwer Nash
So….
Warwick, March 26 2007 17
game over?
Warwick, March 26 2007 18
What next?
efficiency
existence
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 2s-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
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])
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
0
2 3
120 > 10230 > 20310 > 30
oscillation!
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?