Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos”

Computing Equilibria

Christos H. Papadimitriou

UC Berkeley

“christos”

Khachiyan lecture, March 2 06

1,-1 -1,1

-1,1 1,-1

4,4 1,5

5,1 0,0

3,3 0,4

4,0 1,1

matching pennies prisoner’s dilemmachicken

Games help us understand rational behavior in competitive situations


Concepts of rationality

• Nash equilibrium (or double best response)

• Problem: may not exist

• Idea: randomized Nash equilibrium

Theorem [Nash 1951]: Always exists....


can it be found in polynomial time?


is it then NP-complete?

No, because a solution always exists


…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…”


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]


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)


An Easier Problem:Correlated equilibrium

4,4 1,5

5,1 0,0

Chicken:

•Two pure equilibria {me, you}

•Mixed (½, ½) (½, ½) payoff 5/2


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

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


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':

( ) [ ( ) ( ')] 0 i is

x s u s u s


Linear programming!

• n players, s strategies each

• ns2 inequalites

• sn variables!

• Nice for 2 or 3 players

• But many players?


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


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


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, …


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


U 0x max ix

0x TU 1y

0y

need to show dual is infeasible

show it is unbounded


Lemma [Hart and Schmeidler, 89]:For every y there is an x such that

xUTy = 0

•and in fact, x is the product of thesteady-state distributions of the Markovchains implied by y

•Idea: run “ellipsoid against hope”


Leonid Khachiyan [1953-2005]


1 0TxU y

2 0Tx U y

... 0Tkx U y

These k inequalities are themselves infeasible!


TU 1y

0y infeasible

TXU 1y

0y also infeasible

UXT 0x

0x just need to solve


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.


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


Corollaries:

• Graphical games (on any graph!)

• Polymatrix games

• Hypergraphical games

• Congestion games and local effect games

• Facility location games

• Anonymous games

• Etc…


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


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


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!


“Multiplicationis the name of the gameand each generationplays the same…”


The multiplication game

x

y

z = x · y

“affects”

w0 0

0 1

if w plays 0,then it gets xy.if it plays 1,then it gets z,but z gets punished

z wins whenit plays 1 and w plays 0


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


Finally, 4 players

• Previous reduction creates a bipartite graph of degree 3

• Carefully simulate each side by two players, refining the previous reduction


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 equivalentTheorem (with Costas Daskalakis andPaul Goldberg, 2005): …and PPAD-complete


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


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]


In November…

• Conjecture 1: 3-player Nash is also PPAD-complete

• Proved!! [Chen&Deng05, DP05]


In December…

• Conjecture 2: 2-player Nash is in P• PPAD-complete [Chen&Deng05b]


game over!


Thank You!

Computing Equilibria Christos H. Papadimitriou UC Berkeley “christos”

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