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Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses. Ruta Mehta Indian Institute of Technology, Bombay Joint work with Jugal Garg and Albert X. Jiang. A Game: Rock-Paper-Scissor. Rock-Paper-Scissor: A Play. Winner. $ 1. Rock-Paper-Scissor: A Play. Winner. $ 1.

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slide1

Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses

Ruta Mehta

Indian Institute of Technology, Bombay

Joint work with JugalGarg and Albert X. Jiang

bimatrix game
Bimatrix Game

S1 = { R, P, C }

S2 = { R, P, C }

A

B

Steady State: No player gains by unilateral deviation

bimatrix game1
Bimatrix Game

S1 = { R, P, C }

S2 = { R, P, C }

A

B

No Steady State

mixed play
Mixed Play

S1 = { R, P, C }

S1 = { R, P, C }

∆1={r1, p1, c1≥0;

r1+p1+c1=1}

∆2={r2, p2, c2≥0;

r2+p2+c2=1}

B

A

Steady State

john nash 1951
John Nash (1951)
  • Finite Game: Finitely many players, each with finitely many strategies.
  • Nash: Every finite game has a steady state in mixed strategy.

Hence forth called Nash equilibrium (NE)

  • Proved using Kakutani fixed point theorem: Highly non-constructive.
nash equilibrium computation
Nash Equilibrium Computation
  • Papadimitriou (JCSS’94): PPAD-class
    • Problems where existence is guaranteed like fixed point, Sperner’s Lemma, Nash equilibrium.
  • Chen and Deng (FOCS’06): It is PPAD-hard.
  • CDT (FOCS’06): Even approximation is PPAD-hard.
rank and computation
Rank and Computation
  • Kannanand Theobald(SODA’07):
    • Define rank of (A,B) as rank(A+B).
    • FPTAS for fixed rank games.
  • Polynomial time algorithms for exact Nash.
    • Dantzig(1963): Zero-sum (rank-0) is equiv. to LP.
    • AGMS (STOC’11): Rank-1 games.
bilinear games
Bilinear Games

Bimatrix Game with polyhedral strategy sets.

  • Two players: 1and 2
  • Polyhedral strategy sets:
    • X={x | Ex = e; x ≥ 0}, Y={y | Fy=f; y ≥ 0}
  • Payoff matrices: A, B
  • Bilinear Payoff: (x, y) fetches xTAyto player 1, and xTBy to player 2.

Motivation: Koller et al. (STOC’94) for two-player extensive form game with perfect recall.

nash equilibrium in bilinear
Nash Equilibrium in Bilinear
  • NE: No player gains by unilateral deviation.
    • Existence: Corollary of Glicksberg’s result.
  • Symmetric Game:B=AT and Y=X.
    • (x, y) is a symmetric profile if y=x.
    • Existence of symmetric NE: An adaptation of Nash’s proof for symmetric bimatrix games.
bilinear contains
Bilinear Contains:
  • Bimatrix, Polymatrix, Bayesian, etc.
  • Bimatrix: X = ∆1, Y = ∆2
  • Polymatrix:
    • N players. Each pair plays a bimatrix game.
    • Player i: Si finite strategy set, ∆i Mixed strategy set.
    • Goal of i: Choose xi from ∆i to maximize total payoff.

i

Aij

j

polymatrix to bilinear
Polymatrix to Bilinear
  • M= |S1|+ … + |Sn|. X = {(x1,…,xn) | xi in ∆i}, Y=X.
  • A , B=AT

Symmetric NE of (A,B) maps to a NE of the polymatrix game

j

i

A =

best response koller et al
Best Response (Koller et al.)
  • Fix a strategy y of player 2.
  • Player 1 solves

max: xT(Ay) min: eTp

Ex = e pTE≥ (Ay)T

x ≥ 0

At optimal: p s.t. Aiy ≤ pTEi& xi > 0 => Aiy = pTEi

  • Given x X, for player 2 we get

At optimal: q s.t. Bjx ≤ qTFj& yj> 0 => qTFj = Bjx

best response polytopes brps
Best Response Polytopes (BRPs)
  • (x,y) is a NE iff

p: Ay ≤ETp; xi > 0 => Aiy = pTEi

q: xTB≤qTF; yj> 0 => qTFj = Bjx

xT(Ay - ETp) ≤ 0 and (xTB - qTF)y ≤ 0

xT(A+B)y – eTp – fTy ≤ 0

nash equilibrium in brps
Nash Equilibrium in BRPs

NE iffxT(Ay - ETp)=0 and (xTB - qTF)y=0

xT(A+B)y – eTp – fTy=0

Assumption: P and Q are non-degnerate.

(u, v) of P x Q gives a NE => (u, v) is a vertex.

qp formulation
QP Formulation

max: xT(A+B)y – eTp – fTy

s.t. (y, p) P

(x, q) Q

  • Optimal value 0.
  • Only vertex solutions.
our results
Our Results
  • Rank-1 games: rank(A+B)=1
    • Extend Adsul et al. algorithm for exact NE.
  • Fixed rank games: rank(A+B)=k
    • Extend FPTAS of Kannan et al.
  • Rank of A or B is constant
    • Enumerate all NE in polynomial time.
rank 1 case
Rank-1 Case
  • Zero-sum ~ rank(A+B)=0: LP formulation (Charnes’53)
  • rank(A+B)=1 then A+B = a.bT
  • The QP formulation:

max: (xTa)(bTy) – eTp – fTy

s.t. (y, p) P

(x, q) Q

rank 1 case1
Rank-1 Case
  • Replace (xTa) by z. Recall B = -A + a.bT

xT(A+B)y – eTp – fTy=0 z(bTy) – eTp – fTy=0

  • N = Points of P x Q’ with z(bTy) – eTp – fTy=0
    • Forms paths and cycles, since z gives one degree of freedom.

NE of (A,B): Points in intersection of N and z – xTa =0.

parameterized lp
Parameterized LP

LP(z) = max: z(bTy) – eTp – fTy

s.t. (y, p) P

(x, z, q) Q’

  • Given any c, Optimal value of LP(c) is 0.
    • OPT(c) lies on N, and
    • Let N(c)={Points of N with z=c}, then OPT(c)=N(c).
  • N is a single path on which z is monotonic.
rank 1 the algorithm
Rank-1: The Algorithm
  • NE: Intersection of N and H: z – xTa =0.
  • . c1=amin, c2=amax

H

N(c1)

N

H+

H–

NE

N(c2)

rank 1 binary search algorithm
Rank-1: Binary Search Algorithm
  • NE of (A,B): Points in intersection of N and H.
  • c=c1+c2/2.

H

N(c1)

N

H+

H–

NE

N(c)

N(c2)

rank 1 binary search algorithm1
Rank-1: Binary Search Algorithm
  • NE of (A,B): Points in intersection of N and H.
  • c=c1+c2/2. If N(c) in H–,then c1=c else c2=c.

H

N

H+

H–

NE

N(c1)

N(c2)

analysis
Analysis
  • Terminates because,
    • z is monotonic on N.
    • Increase in z on each edge is lower bounded by 1/d where d is polynomial sized in the input.
  • Time complexity:
    • Solve LP(c) to get N(c) in each pivot.
    • log(d) * log(amax – amin) pivots.
conclusions
Conclusions
  • Bilinear games:
    • Bimatrix with polytopal strategy sets.
    • Fairly general. Contains polymatrix, bayesian, etc.
    • Polynomial time algorithm for rank based subclasses.
  • Open problems:
    • Designing a Lemke-Howson type algorithm.
    • Degree, index, stability concepts.
    • Computation of approximate equilibrium.