Welfare-maximizing correlated equilibria using Kantorovich polynomials with sparsity

We provide motivations for the correlated equilibrium solution concept from the game-theoretic and optimization perspectives. We then propose an algorithm that computes <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\varepsilon}$$</EquationSource> </InlineEquation> -correlated equilibria with global-optimal (i.e., maximum) expected social welfare for normal form polynomial games. We derive an infinite dimensional formulation of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\varepsilon}$$</EquationSource> </InlineEquation> -correlated equilibria using Kantorovich polynomials, and re-express it as a polynomial positivity constraint. We exploit polynomial sparsity to achieve a leaner problem formulation involving sum-of-squares constraints. By solving a sequence of semidefinite programming relaxations of the problem, our algorithm converges to a global-optimal <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\varepsilon}$$</EquationSource> </InlineEquation> -correlated equilibrium. The paper ends with two numerical examples involving a two-player polynomial game, and a wireless game with two mutually-interfering communication links. Copyright Springer Science+Business Media, LLC. 2013