Computing Equilibria
Christos H. Papadimitriou
UC Berkeley
“christos”

Games help us understand rational behavior in competitive situations matching pennies
1 , -1 -1 , 1
-1 , 1 1 , -1 chicken
4 , 4 1 , 5
5 , 1 0 , 0 prisoner’s dilemma
3 , 3 0 , 4
4 , 0 1 , 1
Khachiyan lecture, March 2 06

Concepts of rationality
• Nash equilibrium (or double best response)
• Problem: may not exist
• Idea: randomized Nash equilibrium
Theorem [Nash 1951]: Always exists.
.
.
.
Khachiyan lecture, March 2 06

Khachiyan lecture, March 2 06

Khachiyan lecture, March 2 06

…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…”
Khachiyan lecture, March 2 06

Complexity of Nash Equilibria?
• 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]
Khachiyan lecture, March 2 06

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)
Khachiyan lecture, March 2 06

An Easier Problem:
Correlated equilibrium
Chicken:
4 , 4 1 , 5
•Two pure equilibria {me, you}
5 , 1 0 , 0
•Mixed (½, ½) (½, ½) payoff 5/2
Khachiyan lecture, March 2 06

Idea (Aumann 1974)
• “Traffic signal” with payoff 3
• Compare with
Nash equilibrium
0 ½
½
0
1/4 1/4
1/4 1/4
• Even better with payoff 3
1/3
1/3 1/3
1/3 0
Khachiyan lecture, March 2 06
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
Khachiyan lecture, March 2 06

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' :
 s x s u s i
 u s i
( ')]

0
Khachiyan lecture, March 2 06

Linear programming!
• n players, s strategies each
• ns 2 inequalites
• s n variables!
• Nice for 2 or 3 players
•
But many players?
Khachiyan lecture, March 2 06

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 ns n numbers
Khachiyan lecture, March 2 06

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.
2.
multi-player succinctly representable
Khachiyan lecture, March 2 06

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 , ns d numbers suffice to describe the game
• Also: multimatrix, congestion, location, anonymous, hypergraphical, …
Khachiyan lecture, March 2 06

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
Khachiyan lecture, March 2 06

max
 x i
U x

0 show it is unbounded
0
U
T y 1 y

0 need to show dual is infeasible
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
Lemma [Hart and Schmeidler, 89]:
For every y there is an x such that xU T y = 0
•and in fact, x is the product of the steady-state distributions of the Markov chains implied by y
•Idea: run “ellipsoid against hope”
Khachiyan lecture, March 2 06

Leonid Khachiyan [1953-2005]
Khachiyan lecture, March 2 06

x U y

2
T
0
...
T x U y
 k
0 x U y

1
T
0
These k inequalities are themselves infeasible!
Khachiyan lecture, March 2 06

U
T y 1 y

0 infeasible
XU
T y

0 also infeasible
1 just need to solve
Khachiyan lecture, March 2 06
UX T x

0
0

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.
Khachiyan lecture, March 2 06

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
Khachiyan lecture, March 2 06

Corollaries:
• Graphical games (on any graph!)
• Polymatrix games
• Hypergraphical games
• Congestion games and local effect games
• Facility location games
• Anonymous games
• Etc…
Khachiyan lecture, March 2 06

Back to 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
Khachiyan lecture, March 2 06

From d -graphical games to d 2 -normal-form games
• Color the graph with d 2 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
Khachiyan lecture, March 2 06

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!
Khachiyan lecture, March 2 06

“Multiplication is the name of the game and each generation plays the same…”
Khachiyan lecture, March 2 06

x
The multiplication game y
“affects” z wins when it plays 1 and w plays 0 z = x · y w
0 0
0 1
Khachiyan lecture, March 2 06 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
Khachiyan lecture, March 2 06

Finally, 4 players
• Previous reduction creates a bipartite graph of degree 3
•
Carefully simulate each side by two players, refining the previous reduction
Khachiyan lecture, March 2 06

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
Khachiyan lecture, March 2 06

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
Khachiyan lecture, March 2 06

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
Markakis and Mehta 2003]
Khachiyan lecture, March 2 06

In November…
•
[Chen&Deng05,
DP05]
Khachiyan lecture, March 2 06

In December…
• Conjecture 2: 2 -player Nash is in P
• PPAD-complete [Chen&Deng05b]
Khachiyan lecture, March 2 06

Khachiyan lecture, March 2 06

Thank You!
Khachiyan lecture, March 2 06