Computing Equilibria
Christos H. Papadimitriou
UC Berkeley
“christos”
Games help us understand
rational behavior
in competitive situations
matching pennies
chicken
prisoner’s dilemma
1,-1 -1,1
0,0
0,1
3,3
0,4
-1,1 1,-1
1,0
-1,-1
4,0
1,1
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Concepts of rationality
• Nash equilibrium (or double best
response)
• Problem: may not exist
• Idea: randomized Nash equilibrium
Theorem [Nash 1951]: Always exists.
.
.
.
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can it be found
in polynomial time?
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is it then NP-complete?
No, because a
solution always exists
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…and why bother?
(a parenthesis)
• Equilibrium concepts provide some of the
most intriguing specimens of problems
• They are notions of rationality, aspiring
models of behavior
• Efficient computability is an important
modeling prerequisite
“if your laptop can’t find it, then neither
can the market…”
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Brouwer  Nash
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Complexity?
• Nash’s existence proof relies on Brouwer’s
fixpoint theorem
• Finding a Brouwer fixpoint is a hard problem
• Not quite NP-complete, but as hard as any
problem that always has an answer can be…
• Technical term: PPAD-complete [P 1991]
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Complexity? (cont.)
• But how about Nash?
• Is it as hard as Brouwer?
• Or are the Brouwer functions constructed in
the proof specialized enough so that
fixpoints can be computed?
(cf contraction maps)
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Correlated equilibrium
Chicken:
4,4
1,5
5,1
0,0
•Two pure equilibria {me, you}
•Mixed (½, ½) (½, ½) payoff 5/2
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Idea (Aumann 1974)
• “Traffic signal”
with payoff 3
• Compare with
Nash equilibrium
• Even better
with payoff 3 1/3
0
½
½
0
1/4
1/4
1/4
1/4
1/3
1/3
1/3
0
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Probabilities
in a lottery
drawn by an
impartial
outsider, and
announced to
each player
separately
Correlated equilibria
• Always exist (Nash equilibria are
examples)
• Can be found (and optimized over)
efficiently by linear programming
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Linear programming?
• A variable x(s) for each box s
• Each player does not want to deviate
from the signal’s recommendation –
assuming that the others will play along
• For every player i and any two rows of
boxes s, s':
 x(s) [u (s)  u (s ')]  0
i
s
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i
Linear programming!
•
•
•
•
•
n players, s strategies each
ns2 inequalites
sn variables!
Nice for 2 or 3 players
But many players?
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The embarrassing subject
of many players
•
•
•
With games we are supposed to model
markets and the Internet
These have many players
To describe a game with n players and s
strategies per player you need nsn numbers
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The embarrassing subject
of many players (cont.)
•
These important games cannot require
astronomically long descriptions
“if your problem is important, then its
input cannot be astronomically long…”
• Conclusion: Many interesting games are
1.
multi-player
2.
succinctly representable
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e.g., Graphical Games
• [Kearns et al. 2002] Players are vertices of a
graph, each player is affected only by
his/her neighbors
• If degrees are bounded by d, nsd numbers
suffice to describe the game
• Also: multimatrix, congestion, location,
anonymous, hypergraphical, …
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Surprise!
Theorem: A correlated equilibrium in a
succinct game can be found in polynomial
time provided the utility expectation over
mixed strategies can be computed in
polynomial time.
Corollaries: All succinct games in the
literature
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max  xi
U
x  0
x0
show it is unbounded
U  y  1
T
y0
need to show dual is infeasible
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
Lemma [Hart and Schmeidler, 89]:
yx : xU y  0
T
•and in fact, x is the product of the
steady-state distributions of the Markov
chains implied by y
•hence: run “ellipsoid against hope”
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x2U y  0
T
...xkU y  0
T
x1U y  0
T
These k inequalities are themselves infeasible!
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U  y  1
T
XU  y  1
T
y0
y0
infeasible
also infeasible
UXT x  0
just need to solve
x0
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as long as we can solve…
given a succinct representation of a game,
and a product distribution x,
find the expected utility of a player,
in polynomial time.
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And it so happens that…
…in all known cases,
this problem can be solved
by applying one, two, or all three
of the following tricks:
• Explicit enumeration
• Dynamic programming
• Linearity of expectation
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Corollaries:
•
•
•
•
•
•
•
Graphical games (on any graph!)
Polymatrix games
Hypergraphical games
Congestion games and local effect games
Facility location games
Anonymous games
Etc…
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Nash complexity, summary
2-Nash  3-Nash  4-Nash  …  k-Nash  …
|||
1-GrNash  2-GrNash  3-GrNash  …  d-GrNash  …
Theorem (with Paul Goldberg, 2005):
All these problems are equivalent
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From d-graphical games to
d2-normal-form games
• Color the graph with d2 colors
• No two vertices affecting the same vertex
have the same color
• Each color class is represented by a single
player who randomizes among vertices,
strategies
• So that vertices are not “neglected:”
generalized matching pennies
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From k-normal-form games
to graphical games
• Idea: construct special, very expressive
graphical games
• Our vertices will have 2 strategies each
• Mixed strategy = a number in [0,1]
(= probability vertex plays strategy 1)
• Basic trick: Games that do arithmetic!
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“Multiplication
is the name of the game
and each generation
plays the same…”
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The multiplication game
x
z wins when
it plays 1
and w plays 0
“affects”
z=x·y
w
y
0
0
0
1
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if w plays 0,
then it gets xy.
if it plays 1,
then it gets z,
but z gets punished
From k-normal-form games
to 3-graphical games (cont.)
• At any Nash equilibrium, z = x y
• Similarly for +, -, “brittle comparison”
• Construct graphical game that checks the
equilibrium conditions of the normal form
game
• Nash equilibria in the two games coincide
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Finally, 4 players
• Previous reduction creates a bipartite graph
of degree 3
• Carefully simulate each side by two
players, refining the previous reduction
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Nash complexity, summary
2-Nash  3-Nash  4-Nash  …  k-Nash  …
|||
1-GrNash  2-GrNash  3-GrNash  …  d-GrNash  …
Theorem (with Paul Goldberg, 2005):
All these problems are equivalent
Theorem (with Costas Daskalakis and
Paul Goldberg, 2005): …and PPAD-complete
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Nash is PPAD-complete
• Proof idea: Start from a PPAD-complete
stylized version of Brouwer on the 3D cube
• Use arithmetic games to compute Brouwer
functions
• Brittle comparator problem solved by
averaging
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Recall:
• Nash’s theorem reduces Nash to
Brouwer
• This is a reduction in the opposite
direction
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So….
Brouwer  Nash
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Open problems
• Conjecture 1: 3-player Nash is also
PPAD-complete
• Conjecture 2: 2-player Nash can be
found in polynomial time
• Approximate equilibria? [cf. Lipton
and Markakis 2003]
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In November…
• Conjecture 1: 3-player Nash
is also PPAD-complete
• Proved!! [Chen&Deng05,
DP05]
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In December…
• Conjecture 2: 2-player Nash is in P
• PPAD-complete [Chen&Deng05b]
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game over!
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