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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
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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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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
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Equilibria: the trade-offs
mixed Nash
[DGP06, CD06]
correlated
existence
efficiency
naturalness
pure Nash
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Equilibria: the succinct case
mixed Nash
[DFP ICALP06]
correlated
[PR SODA-STOC05]
existence
efficiency
naturalness
pure Nash
NP-c/PLS-c [FPT03]
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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 graph
with indegree, outdegree < 2
Standard source
(given)
?
(there must
be 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
“affects”
z=x·y
w
y
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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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So….
Brouwer  Nash
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game over?
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What next?
?
existence
efficiency
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 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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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
1
2
A strategy for the row player
a
b
1
2
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])
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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
120 > 10
230 > 20
310 > 30

oscillation!
0
2
3
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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?
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