Simple search methods for finding a Nash equilibrium
Ryan Porter, Eugene Nudelman, and Yoav Shoham
Games and Economic Behavior, Vol. 63, Issue 2. pp. 642-661, 2004
Georgia Kastidou
David R. Cheriton School of Computer Science
University of Waterloo
Outline
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Problem
Contribution
Algorithm for 2-player game
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Algorithm for n-player game
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Experimental Results
Experimental Results
Conclusions
Future Work
Take Home Message
Problem we will tackle…
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Consider a n-player normal-form game G=N, (Ai), (ui) where:

N=[1,..n] is the set of players
Ai=[ai1,…,aimi] is the set of actions available to player i
ui:A1x…xAnR is the utility function for each player
Each player selects a mixed strategy from the set of available strategies:


Pi   pi : Ai  [0,1] |  pi (ai )  1
ai  Ai


“Support” of a mixed strategy pi is the set of all actions ai in Ai such that pi(ai)>0.
x=(x1,..,xn): xi is the size of agent’s i support

The expected utility for player i for a strategy profile p=(p1,...,pn) is:





where:
ui ( p) 
 p (a)u (a)
a A
i
i
pi (a)   pi (ai )
iN
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Problem: Design an algorithm which can find a Nash Equilibrium for a
normal form game

A strategy profile p* in P is a Nash Equilibrium (NE) if:
i  N , ai  Ai :
ui (ai , p*i )  ui ( pi* , p*i )
Development of Algorithms
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Two extremes:
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1st: Aims to design a low complexity algorithm
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2nd: Aims to design more simple algorithms with an “acceptable”
complexity
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Gain deep insight in the structure of the problem
Design highly specialized algorithms
Identify shallow heuristics
Hoping that given the increasing computing power they will be
sufficient.
focus on more “common” problems
Which one is better?
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Although the first is more interesting, in practice in a number of
cases the second is preferred either because is simpler or
because it outperforms the first.

Note: There are cases of optimal algorithms that have never been implemented because
they are too complicate.
Background/Related Work

Nash equilibrium is an important concept in game
theory

“little is known about the problem of computing a sample
NE in a normal-form game”
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Normal form game is guaranteed to have at least
one NE
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It does not fall into a standard complexity class
(Papadimitriou, 2001)

it cannot be cast as a decision problem.
Background/Related Work
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2-player game:
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Lemke-Howson, (1964)
Dickhaut and Kaplan (1991)
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Enumerates all possible pairs of supports for 2-play game
For each pair it solves a feasibility program
n-player game:

Simplicial Subdivision (van der Laan et al., 1987)
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
Approximates a fixed point of a function which is defined on a simplotope
Govindan and Wilson (2003)


first perturbs a game to one that has a known equilibrium, and
then traces the solution back to the original game as the magnitude of the
perturbation approaches zero.
Background/Related Work

GAMUT: a “recently” (2004) introduced
computational testbed for game theory
“Run the GAMUT:A Comprehensive Approach to Evaluating
Game-Theoretic Algorithms” by E. Nudelman et al.
What the author propose and
what’s their contribution?

Propose:
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heuristic-based algorithms for 2-player games and for
n-player games
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
test using a variety of different distributions
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explore the space of support profiles using a backtracking procedure
to instantiate the support for each player separately.
Use of GAMUT (computational testbed for game theory)
Contribution:

in a big number of cases the proposed algorithms
outperform the algorithm of Lemke-Howson on 2player games and the Siplicial Subdivision on n-player
games.
The Proposed Algorithms
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explore “support” profiles:
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pure strategies played with nonzero probability
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use backtracking procedures
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are biased towards simple solutions
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preference for small supports
based on the observation that a number of
games in the past proved to have at least one
simple solution.
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e.g. for n = 2, the probability that there exists a NE consistent
with a particular support profile varies inversely with the size
of the supports, and is zero for unbalanced support profiles.
Proposed Algorithm for 2-players game
Proposed Algorithm for 2-players game
υi : expected utility of agent i in an equilibrium
Proposed Algorithm for 2-players game

The first two classes of constraints
require that:
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
each player must be indifferent between
all actions within his/her support, and
must not strictly prefer an action outside
of his/her support.
Experimental results

The authors consider games from a number of different
distributions

D18: most common one
D5, D6, and D7 are also important distributions
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Experimental Results
Algorithms for 2-player games
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Experiment-Setup:


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First diagram:
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
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compares the unconditional median running times of the algorithms,
might reflect the fact that there is a greater than 50% chance that the
distribution will generate a game with a pure
Second diagram:
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
2-player, 300-action games drawn from 24 of GAMUT’s 2- player
distributions.
executed on 100 games drawn from each distribution.
Compares the percentage of instances solved
Third diagram:

the average running time conditional on solving an instance
Experimental Results
Algorithms for 2-player games
Compares the unconditional median running times of the
algorithms.
(“Might reflect the fact that there is a greater than 50% chance
that the distribution will generate a game with a pure”)
Experimental Results
Algorithms for 2-player games
Compares the percentage of instances solved
Experimental Results
Algorithms for 2-player games
Compares the average running time conditional on
solving an instance
(unconditional average running time)
Experimental Result
Algorithms for 2-player games
Compare the scaling behavior as the number of actions increases
(unconditional average running time)
Experimental Result
Algorithms for 2-player games
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Covariance Games
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neither of the algorithms solved any of the games in another
“Covariance Game” distribution in which ρ =−0.9,
Proposed Algorithm for n-players
games
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Uses a general backtracking algorithm to solve a
constraint satisfaction problem (CSP) for each
support size profile
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The variables in each CSP are:



the supports Si , and
the domain of each Si is the set of supports of size xi.
Constraints:

no agent plays a conditionally dominated action.
Proposed Algorithm for n-players
games

IRSDS:

Input a domain for each player’s support.

For each agent whose support has been instantiated the domain
contains only that instantiated support,
For each other agent i it contains all supports of size xi that were not
eliminated in a previous call to this procedure.
On each pass of the repeat-until loop,


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every action found in at least one support in a player’s domain is checked for
conditional domination.
If a domain becomes empty after the removal of a conditionally dominated
action,
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
the current instantiations of the Recursive-Backtracking are inconsistent, and IRSDS
returns failure.
IRSDS repeats until it either returns failure or iterates through all actions
of all players without finding a dominated action.
Proposed Algorithm for n-players
games
Algorithm 2
For all x=(x1,..xn) sorted in
increasing order first by:
x
i
i
R-B
IRSDS
IRSDS
R-B
R-B
IRSDS
IRSDS
R-B
R-B
Failed
…
IRSDS
R-B
… xIRSDS
i
i
R-B
IRSDS
RBT
Failed
and then by:
max i , j ( xi  x j )
R-B: Recursive Backtracking
IRSDS: Iterated Removal of Strictly Dominated Strategies
Experimental Results: n-player games
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Experiment-Setup:
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6-player, 5-action games drawn from 22 of GAMUT’s n-player distributions.
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


compares the unconditional median running times of the algorithms,
might reflect the fact that there is a greater than 50% chance that the
distribution will generate a game with a pure
Second diagram:


executed on 100 games drawn from each distribution.
First diagram:


15,625 outcomes and 93,750 payoffs
Compares the percentage of instances solved
Third diagram:

the average running time conditional on solving an instance
Experimental Results: n-player games
Compares the unconditional median
running times of the algorithms
Compares the percentage of instances
solved
Experimental Results: n-player games
Compares the average running time
conditional on solving an instance
Experimental Results: n-player games
Compare the scaling behavior: number of
players constant at 6 number of actions
varies.
Compare the scaling behavior: number of
players varies, number of actions
constant 5.
(unconditional average running time)
(unconditional average running time)
Experimental Results
Percentage of Pure Strategy NE
(2-player game)
n-player game
Experimentals Results
Average measure of support balanced
2-player, 300-action games
6-player, 5-action games
Conclusions

Propose algorithms that use backtracking
approaches to search the space of support profiles,
favoring supports that are small and balanced.

Both algorithms outperform the current state of the
art.

The most difficult games


“Covariance Game” model, as the covariance approaches
its minimal value
hard because authors found that:
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
as the covariance decreases, the number of equilibria
decreases, and
the equilibria that do exist are more likely to have support
sizes near one half of the number of actions
Future Work

Employ more sophisticated CSP techniques

Explore local search, in which the state space is the
set of all possible supports, and the available
moves are to add or delete an action from the
support of a player

Study the games that are generated by the
Covariance Game distribution
Take home message

Studying the results of complicated problems can
lead to observations that although might not provide
ideas to find optimal solutions can provide insights
on how to improve current approaches.

The selection of the tests and the parameters that
will be examined very important.


Not only because they can show that your algorithm is
working…
E.g. “Covariance Game” model might proved a good starting
point for game theoretic algorithms