A Projection Framework for Near-
Potential Polynomial Games
IEEE CDC Maui, December 13 th 2012
Nikolai Matni
( nmatni@caltech.edu
)
Control and Dynamical Systems,
California Institute of Technology

• Informal definition: local actions have predictable global consequences.
• Nice properties
– Pure-strategy Nash Equilibria (NE)
– Simple dynamics converge to these NE
• Applications to distributed control
– Marden, Arslan & Shamma 2010
– Candogan, Menache, Ozdaglar
& Parrilo 2009
– Li & Marden, 2011

• Would like to consider general class of continuous games
– Finite players, continuous action sets.
• Why?
– Goal is control: most systems of interest are analog.
– Quantization leads to tradeoffs in granularity, performance and problem dimension.
• Why not?
– Potentially intractable to analyze (Parrilo 2006, Stein et al. 2006 for recent results).
– Can lead to infinite dimensional optimization problems.
• Solution?
– Restrict ourselves to polynomial cost functions and use Sum Of
Squares (SOS) methods.

• O. Candogan, A. Ozdalgar, P.A. Parrilo, A Projection
Framework for Near-Potential Games, CDC 2010 (and subsequent work)
• Basic idea: if a game is “close” to being a potential game, it behaves “almost as well.”
• Projection Framework – finite dimensional case
– Potential games form a subspace.
– Project onto this framework to find closest potential game.
– If distance from subspace is small, original game inherits many nice properties.
• Goal: Extend these ideas to polynomial games.

• Motivation
– Potential games
– Polynomial games
– Near-Potential games
• Preliminaries
– Game Theory
– Algebraic Geometry/Sum of Squares (SOS)
• Projection Framework
• Properties
– Static
– Dynamic
• Example
• Conclusions and Future work

• Motivation
– Potential games
– Polynomial games
– Near-Potential games
• Preliminaries
– Game Theory
– Algebraic Geometry/Sum of Squares (SOS)
• Projection Framework
• Properties
– Static
– Dynamic
• Example
• Conclusions and Future work

• A polynomial game
– A finite player set
– Strategy spaces
– Polynomial utility functions
,
, where is given by:
• A polynomial game is:
– Continuous if for all n, is a closed interval of the real line
– Discrete if for all n,
– Mixed if some strategy sets are continuous, and some are discrete.
– Assume w.l.o.g.

• A polynomial game G is a polynomial potential
game if there exists a polynomial potential function player n, and every such that, for every
• Algebraic characterization (Monderer, Shapley
’96): A continuous game is a potential game iff

• A strategy is an approximate Nash (or
ε) Equilibrium if, for all n, we have that

• Definition: a real polynomial p(x) admits a Sum Of Squares
(SOS) decomposition if
• Why SOS?
– Determining if p(x)≥0, is in general, NP-hard
– Determining if p(x) is SOS tested through SDP
• Lemma [SOS relaxation]:
If there exist SOS polynomials such that then

• Motivation
– Potential games
– Polynomial games
– Near-Potential games
• Preliminaries
– Game Theory
– Algebraic Geometry/Sum of Squares (SOS)
• Projection Framework
• Properties
– Static
– Dynamic
• Example
• Conclusions and Future work

Projection Framework – MPD & MDD
• Need a notion of distance in the space of games
• Candogan et al. introduced Maximum Pairwise
Distance (MPD)
• Use the continuity of polynomials to define Maximum
Differential Difference (MDD)
• Both capture how different two games are in terms of utility improvements due to unilateral deviations

• Task: Given a polynomial game , find a nearby potential polynomial game
• Formulate as an optimization problem:
• Constraint ensures we get a Potential Game
• Objective function minimizes MDD.
• Intractable!

• Step 1: rewrite constraint in terms of algebraic characterization
• Step 2: introduce slack variable γ

• Step 3: apply Lemma [SOS relaxation]
• This is a finite dimensional SOS program, solvable in polynomial time. It yields a polynomial potential game satisfying

• Can extend this idea to mixed/discrete games
• Lemma [MPD]:
If , then
• Continuous Relaxations: For a mixed or discrete game, set all strategy sets to [-1,1]
– Apply previous SOS program and Lemma [MPD] to mixed games or discrete games with
– Allows us to apply algebraic characterization, which can reduce number of constraints from O( ) to O(N)

• Motivation
– Potential games
– Polynomial games
– Near-Potential games
• Preliminaries
– Game Theory
– Algebraic Geometry/Sum of Squares (SOS)
• Projection Framework
• Properties
– Static
– Dynamic
• Example
• Conclusions and Future work

• Let and be such that .
Then for every ε
1
-equilibrium y of , z(y) is an
ε-equilibrium of , where
• For continuous games, D=0, z(y)=y, and local maxima of P are pure (ε=0) NE.

• Let and be such that .
Then for every ε
1
-equilibrium y of , z(y) is an
ε-equilibrium of , where
• For continuous games, D=0, z(y)=y, and local maxima of P are pure (ε=0) NE.

• Definition: ε-better response dynamics
– Round robin updates
– Player updates only to improve utility by at least ε
– Otherwise does not update
• Suppose there exists such that
Then, under ε-better response dynamics, after a finite number of iterations, dynamics will be
confined to the ε-equilibria set of , for arbitrary.

• Motivation
– Potential games
– Polynomial games
– Near-Potential games
• Preliminaries
– Game Theory
– Algebraic Geometry/Sum of Squares (SOS)
• Projection Framework
• Properties
– Static
– Dynamic
• Example
• Conclusions and Future work

• Consider the N player game defined by
–
–
–
• Distributed power minimization interpretation

• Run through projection framework to find nearby potential game : satisfying

• Potential function concave – can compute global maximum to identify .2-equilibria of G
• Alternatively, can run .2-better response dynamics to converge to a .2-equilibria of G.
• Quantify performance through cost function

• Compare better-response
(x x br
) to centralized (optimal
*
) positions
• Better response comes within ~20% of centralized solution
• Completely decentralized
• Arbitrarily scalable
• Requires no a priori knowledge of base station locations

• Introduce framework for analyzing polynomial games
– Defined MDD and a tractable projection framework to find nearby potential games
– Related static and dynamic properties of polynomial games to those of nearby potential games
– Illustrated these methods on a distributed power problem
• Future work
– Projecting onto weighted polynomial games
– Additional static properties (mixed-equilibria)
– Efficiency notions (price of anarchy, price of stability, etc.)