Download presentation

Presentation is loading. Please wait.

Published byColten Killman Modified over 2 years ago

1
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

2
A Game: Rock-Paper-Scissor

3
Rock-Paper-Scissor: A Play Winner $1$1

4
Rock-Paper-Scissor: A Play Winner $1$1

5
Rock-Paper-Scissor: A Play Winner $1$1

6
0,0-1,11,-1 0,0-1,1 1,-10,0 Rock-Paper-Scissor Payoffs

7
RPC R01 P10 C 10 Bimatrix Game Steady State: No player gains by unilateral deviation RPC R01 P 01 C1 0 S 1 = { R, P, C } S 2 = { R, P, C } AB

8
RPC R01 P10 C 10 Bimatrix Game No Steady State RPC R01 P 01 C1 0 S 1 = { R, P, C } S 2 = { R, P, C } AB

9
R 1/3 P 1/3 C 1/3 R01 P10 C 10 Mixed Play Steady State RPC R 1/301 P 1/301 C 1/310 S 1 = { R, P, C } A B ∆ 1 ={r 1, p 1, c 1 ≥0; r 1 +p 1 +c 1 =1} S 1 = { R, P, C } ∆ 2 ={r 2, p 2, c 2 ≥0; r 2 +p 2 +c 2 =1}

10
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.

11
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.

12
Rank and Computation Kannan and 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.

13
Bilinear Games Bimatrix Game with polyhedral strategy sets. Two players: 1 and 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 x T Ay to player 1, and x T By to player 2. Motivation: Koller et al. (STOC’94) for two-player extensive form game with perfect recall.

14
Nash Equilibrium in Bilinear NE: No player gains by unilateral deviation. Existence: Corollary of Glicksberg’s result. Symmetric Game: B=A T 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.

15
Bilinear Contains: Bimatrix, Polymatrix, Bayesian, etc. Bimatrix: X = ∆ 1, Y = ∆ 2 Polymatrix: N players. Each pair plays a bimatrix game. Player i: S i finite strategy set, ∆ i Mixed strategy set. Goal of i: Choose x i from ∆ i to maximize total payoff. A ij i j

16
Polymatrix to Bilinear M= |S 1 |+ … + |S n |. X = {(x 1,…,x n ) | x i in ∆ i }, Y=X. A, B=A T Symmetric NE of (A,B) maps to a NE of the polymatrix game 0 0 A ij 0 0 i j A =

17
Best Response (Koller et al.) Fix a strategy y of player 2. Player 1 solves max: x T (Ay) min: e T p Ex = e p T E ≥ (Ay) T x ≥ 0 At optimal: p s.t. A i y ≤ p T E i & x i > 0 => A i y = p T E i Given x X, for player 2 we get At optimal: q s.t. B j x ≤ q T F j & y j > 0 => q T F j = B j x

18
Best Response Polytopes (BRPs) (x,y) is a NE iff p: Ay ≤ E T p; x i > 0 => A i y = p T E i q: x T B ≤ q T F; y j > 0 => q T F j = B j x x T (Ay - E T p) ≤ 0 and (x T B - q T F)y ≤ 0 x T (A+B)y – e T p – f T y ≤ 0

19
Nash Equilibrium in BRPs NE iff x T (Ay - E T p)=0 and (x T B - q T F)y=0 x T (A+B)y – e T p – f T y=0 Assumption: P and Q are non-degnerate. (u, v) of P x Q gives a NE => (u, v) is a vertex.

20
QP Formulation max: x T (A+B)y – e T p – f T y s.t.(y, p) P (x, q) Q Optimal value 0. Only vertex solutions.

21
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.

22
Rank-1 Case Zero-sum ~ rank(A+B)=0: LP formulation (Charnes’53) rank(A+B)=1 then A+B = a. b T The QP formulation: max: (x T a)(b T y) – e T p – f T y s.t.(y, p) P (x, q) Q

23
Rank-1 Case Replace (x T a) by z. Recall B = -A + a. b T x T (A+B)y – e T p – f T y=0 z(b T y) – e T p – f T y=0 N = Points of P x Q’ with z(b T y) – e T p – f T y=0 Forms paths and cycles, since z gives one degree of freedom. NE of (A,B): Points in intersection of N and z – x T a =0.

24
Parameterized LP LP(z) = max: z(b T y) – e T p – f T y 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.

25
Rank-1: The Algorithm NE: Intersection of N and H: z – x T a =0. . c 1 =a min, c 2 =a max H N H–H– H+H+ NE N (c 1 ) N (c 2 )

26
Rank-1: Binary Search Algorithm NE of (A,B): Points in intersection of N and H. c=c 1 +c 2 /2. H NE N (c 1 ) N (c 2 ) N N (c) H+H+ H–H–

27
Rank-1: Binary Search Algorithm NE of (A,B): Points in intersection of N and H. c=c 1 +c 2 /2. If N (c) in H –,then c 1 =c else c 2 =c. H NE N (c 2 ) N N (c 1 ) H+H+ H–H–

28
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(a max – a min ) pivots.

29
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.

30
Thank You

Similar presentations

OK

Small clique detection and approximate Nash equilibria Danny Vilenchik UCLA Joint work with Lorenz Minder.

Small clique detection and approximate Nash equilibria Danny Vilenchik UCLA Joint work with Lorenz Minder.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on tata group of companies Ppt on non renewable sources of energy Ppt on tata trucks commercial vehicle Ppt on microsoft project plan Difference between lcd and led display ppt on tv Ppt on switching devices on at&t Ppt on adr and drive Ppt on art of war author Ppt on mutual fund industry in india Ppt on field study 2