1. Simple Search Methods for Finding a Nash Equilibrium
3/25/2017
Simple Search Methods for Finding a Nash Equilibrium
Ryan Porter, Eugene Nudelman, & Yoav Shoham
Computer Science Department
Stanford University

2. Finding a Sample Nash Equilibrium
3/25/2017
Finding a Sample Nash Equilibrium
Nash equilibrium (NE)
Arguably the most important concept in game theory
One always exists [N51]
Finding a sample NE in a normal form game:
Considered hard, but unknown whether it is NP-hard
State of the art among existing algorithms:
Lemke-Howson [LH64]
Simplicial Subdivision [VV87]
Govindan-Wilson [GW03] & [BSK03]
Our algorithms: simple Artificial Intelligence methods that perform well in practice
2-player games
“...most fundamental computational problem whose complexity is wide open” [P01]
N-player games

3. Notation Normal Form Game G = h N,(Ai),(ui) i:
3/25/2017
Notation
Normal Form Game G = h N,(Ai),(ui) i:
N = {1,…,n}: set of players
Ai: set of available actions for player i
ui: A1 x …x An ! <
Player i selects a mixed strategy:
pi: Ai ! [0,1], s.t. ai 2 Ai pi(ai) = 1
Utility function extended to take p=(p1,…,pn):
ui(p) = a 2 A ui(a) i 2 N pi(ai)
A strategy profile p* is a NE if:
8 i 2 N, ai 2 Ai: ui(ai,p*-i) ≤ ui(p*i,p*-i)
1/2
1/2
1,-1
-1,1
1/2
1/2

4. 3/25/2017
A Harder Game
2/7
3/7
2/7
2,3
-1,4
2,4
5,2
1,-1
2,2
3,0
4,1
-2,4
1,3
4,6
7,2
2,-2
4,9
2,1
9,0
-2,6
6,3
7,0
0,5
3,2
6,1
2,5
5,3
1,0
2/11
4/11
The problem of finding a NE can be formulated as a complementarity problem, which, even in the 2-player case, is hard
5/11

5. Searching Over Supports
3/25/2017
Searching Over Supports
Feasibility Problem:
Input: S = (S1,...,SN), where 8 i 2 N, Si µ Ai
Find: p=(p1,…,pn) and v=(v1,…,vn)
Subject to:
8 i 2 N
8 ai 2 Si, pi(ai) ≥ 0
8 ai 2 Si, pi(ai) = 0
ai 2 Ai pi(ai) = 1
8 ai 2 Si, a-i 2 A-i ui(ai,a-i) ji p(aj) = vi
8 ai 2 Si, a-i 2 A-i ui(ai,a-i) ji p(aj) ≤ vi
2/7
3/7
2/7
2,3
-1,4
2,4
5,2
1,-1
2,2
3,0
4,1
-2,4
1,3
4,6
7,2
2,-2
4,9
2,1
9,0
-2,6
6,3
7,0
0,5
3,2
6,1
2,5
5,3
1,0 2 2
2/11 4 4
4/11
5/11 4 2 3 3 3 2

6. Features of Algorithm Prefer balanced supports Prefer small supports
3/25/2017
Features of Algorithm
Prefer balanced supports
Prefer small supports
Motivated by existing theoretical results for particular distributions (e.g., [MB02])
Separately instantiate supports, and remove conditionally dominated actions:
An ai is conditionally dominated, given R-i µ A-i if:
9 ai' 2 Ai, 8 a-i 2 R-i, ui(ai,a-i) < ui(ai',a-i)
Especially useful in conjunction with (2)
The distribution of [MB02] is equivalent to one in which each payoff is independently drawn from a normal distribution
In 2-player games under this model, there is zero probability that a NE is consistent with any particular unbalanced support profile
Probability of existence of NE consistent with support profile varies inversely with the size of the support

7. 3/25/2017
Two-Player Algorithm
FOR ALL x = (x1,x2), sorted in increasing order of
|x1 – x2| and (x1 + x2)
FOR ALL S1 µ A1 s.t. |S1| = x1
A2' ← {a2 2 A2 not conditionally dominated, given S1}
a1 2 S1 conditionally dominated, given A2'
FOR ALL S2 µ A2' s.t. |S2| = x2
a1 2 S1 conditionally dominated, given S2
IF Feasibility Problem satisfied for (S1,S2)
Return found NE p

8. 3/25/2017
N-Player Algorithm
Constraint Satisfaction Problem (CSP) for each support size profile x=(x1,x2):
Variables: Si
Domain: all subsets of Ai of size xi
Constraint: support profile S is consistent with a NE
2-player algorithm:
Backtracking, enforcing arc consistency w.r.t. weaker constraints that no conditionally dominated actions in S
N-player algorithm:
Generalizes the 2-player algorithm
Ordering of size and balance reversed
Second set of constraints is weaker, but easier to check

9. 3/25/2017
Experimental Results
Most previous empirical tests only on “random” games:
Each payoff drawn independently from uniform distribution
GAMUT [NWSL04]
Based on extensive literature search
Generates games from a wide variety of distributions
Available at D1
Bertrand Oligopoly D2
Bidirectional LEG, Complete Graph D3
Bidirectional LEG, Random Graph D4
Bidirectional LEG, Star Graph D5
Covariance Game:  = 0.9 D6
Covariance Game:  = 0 D7
Covariance Game: Random 2 [-1/(N-1),1] D8
Dispersion Game D9
Graphical Game, Random Graph
D10
Graphical Game, Road Graph
D11
Graphical Game, Star Graph
D12
Location Game
D13
Minimum Effort Game
D14
Polymatrix Game, Random Graph
D15
Polymatrix Game, Road Graph
D16
Polymatrix Game, Small-World Graph
D17
Random Game
D18
Traveler’s Dilemma
D19
Uniform LEG, Complete Graph
D20
Uniform LEG, Random Graph
D21
Uniform LEG, Star Graph
D22
War Of Attrition
based on an extensive literature search of economics, game theory, computer science

10. 3/25/2017
2-player Games
Tested on player, 300-action games for each of 22 distributions
Capped all runs at 1800s

11. 2-player Games: Scaling
3/25/2017
2-player Games: Scaling

12. 2-player Games: Covariance Games
3/25/2017
2-player Games: Covariance Games
Covariance Games: For each action profile, payoffs of all players drawn from a multivariate normal distribution, with identical covariance between any two players
1000 points

13. 3/25/2017
N-player Games
Tested on player, 5-action games for each distribution

14. N-player Games: Scaling
3/25/2017
N-player Games: Scaling

15. N-player Games: Covariance Games
3/25/2017
N-player Games: Covariance Games

16. BFS Lemke-Howson Lemke-Howson algorithm:
3/25/2017
BFS Lemke-Howson
Lemke-Howson algorithm:
Pivoting method to solve LCP for a 2-player game
First pivot is an arbitrary selection of a1 2 A1
Afterwards, a deterministic path to a NE
Idea: favor “simple” solutions
Breadth-First Search:
FOR ALL a1 2 A1
Initialize Lemke-Howson(a1)
REPEAT
Pivot Lemke-Howson(a1)
IF found a NE, THEN return p

17. 2-player “Random” Games
3/25/2017
2-player “Random” Games

18. 2-player Games: Covariance Games
3/25/2017
2-player Games: Covariance Games

19. Summary CSP-based algorithms BFS Lemke-Howson Commentary on problem:
3/25/2017
Summary
CSP-based algorithms
Heuristics:
Favor balanced and small supports
Eliminate conditionally dominated strategies
Perform well in practice
BFS Lemke-Howson
In preliminary results, performs even better than our 2-player algorithm
Commentary on problem:
Games researchers care about tend to have at least one “simple” solution

20. Future Work Coming to Gambit
3/25/2017
Future Work
Coming to Gambit
Focus on “Covariance” Games, with low covariance
Other techniques from Artificial Intelligence
Local Search:
State: support profile
Operators: add or delete an action
Score: based on relaxation of the feasibility problem

21. Simple Search Methods for Finding a Nash Equilibrium
3/25/2017
Simple Search Methods for Finding a Nash Equilibrium
Ryan Porter, Eugene Nudelman, & Yoav Shoham
Computer Science Department
Stanford University