A Polynomial Time Algorithm for 2 -Player Rank 1 Games

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A Polynomial Time Algorithm for 2 -Player Rank 1 Games. Ruta Mehta. Based on a joint work with Bharat Adsul , Jugal Garg and Milind Sohoni. m strategies. n strategies. B. A. Mixed Strategy/Randomize. B. A. Goal: Maximize expected payoff. Nash Equilibrium. John Nash (1951):

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A Polynomial Time Algorithm for

2-Player Rank 1 Games

Ruta Mehta

Based on a joint work with

m strategies

n strategies

B

A

Mixed Strategy/Randomize

B

A

Goal: Maximize expected payoff

Nash Equilibrium

John Nash (1951):

Given a finite game, there exists a tuple of mixed-strategy vectors, one for each player, such that no player gains by deviating unilaterally.

Computation

k-Nash: Computing a Nash equilibrium of a finite k player game.

2-Nash:Rational Solutions

DGP’06, CD’06)

k-Nash, : Algebraic Solutions

Computation

k-Nash: Computing a Nash equilibrium of a finite k player game.

2-Nash:Rational Solutions

DGP’06, CD’06)

k-Nash, : Irrational Solutions (Nash’51)

FIXP-complete (Etessami &

Yannakakis’07)

2-Nash
• von Neumann (1928):In zero-sum games (), Min-Max strategies are stable.
• Linear programming duality In P.

• Kannan Theobald (2005): Defined rank of a game as rank(A+B).
• Zero-sum rank 0.
2-Nash
• von Neumann (1928):In zero-sum games (), Min-Max strategies are stable.
• Linear programming duality In P.

• Kannan Theobald (2005): Defined rank of a game as rank(A+B).
• Zero-sum rank 0. FPTAS for fixed-rank.
• Can we solve rank 1 games in polynomial time?
Difficulty in Rank 1 Games

Disconnected set of equilibria(KT’05).

Exponentially many disconnected equilibria (von Stengel’12).

2-Nash

n strategies

m strategies

A

B

• Mixed strategy
• Probability distribution vectors
2-Nash

n strategies

m strategies

A

B

• Mixed strategy

is a Nash equilibriumiff

• fetches the best payoff to Alice, against .
• fetches the best payoff to Bob, against
2-Nash Characterization

n strategies

m strategies

A

B

If Bob plays then for Alice

• Expected payoff from her strategy is

A

Bob plays . Then for Alice

• Expected payoff from her strategy is

A

• Best strategies are
• achieves max payoffiff

Related toComplementarity

For Bob

Related toComplementarity

Fixing for Alice, Bob’s payoff is

from strategy.

Best strategies are

achieves max payoffiff

Define and

Complementarity

Complementarity and

Nash equilibrium

Polyhedra

Variable vector

Scalar variable

Variable vector

Scalar variable

Complementarity:

Vertex

is a NE

are the payoffs

At least the sum of

max payoff

Sum of payoffs

At least the sum of

max payoff

Sum of payoffs

is a NE

are the max payoffs

At least the sum of

max payoff

Sum of payoff

2-Nash

max:

Complementarity

s.t.

is a NE

are the max payoffs

Rank 1 Game

Bilinear

2-Nash

max:

s.t.

Rank 1 Game

Product of two linear terms

2-Nash

max:

s.t.

Rank 1 QP is NP-hard in general

Think Big!

Consider a space of rank 1 games

S = {

Think Big!

Consider a space of rank 1 games

Think Big!

2-Nash

max:

s.t.

Think Big!

Consider game space

2-Nash

max:

s.t.

Think Big!

Consider game space

All NE of S

Complementarity

Idea

Captures

max:

Solutions of LP(

s.t.

LP()

(,), a NE of S

Proof:

At any feasible point, cost of LP() is at most zero.

is feasible.

Complementarity Cost is exactly zero at it.

Claim:

LP()

LP()

Goal: NE of

If one of them

then done!

(m-1)-dimensional

space in S

Claim:s.t.

(,) is a NE of game

Goal: NE of game

LP()

If = then done!

Goal: NE of game

LP()

NE of game Fixed points of

If = then done!

1-D Fixed Point

Lower

Upper

1

Upper

Lower

0

1

1-D Fixed Point

Lower

Upper

1

Upper

Lower

0

1

And so on until the difference becomes small enough

Kohlberg and Mertens (1986)

Finite player

Game space

x

Homeomorphic

Its Nash

correspondence

g

Kohlberg and Mertens (1986)

Finite player

Game space

x

Continuous

Bijection

Its Nash

correspondence

g

Stability, Index (Shapley’74, KM’86, GW’97)

To design homotopy methods. (Govindan & Wilson’03,…)

No extension to subspaces known

We Show That

Game space

S =

Its Nash

correspondence

Homeomorphic

Rank 2 Games

Open: A polynomial time algorithm.

Simplex method: Practical, although

(Dantzig’47) exponential in worst case

In P or NP-hard?

Khachiyan’79: Polynomial time solvable.

vs.

Stumbling

Block