Quadratic Programming Approach for Nash Equilibrium Computation in Multiplayer Imperfect-Information Games
This provides a method for exact equilibrium computation in multiplayer imperfect-information games, addressing a gap where prior scalable algorithms only worked for two-player zero-sum cases, though it is incremental as it builds on existing formulations and focuses on small-scale examples.
The paper tackles the problem of computing exact Nash equilibrium in multiplayer imperfect-information games, which existing scalable methods like counterfactual regret minimization fail to guarantee convergence for. It presents a quadratic programming approach that solves three-player Kuhn poker faster and more accurately than the Gambit software suite's logit quantal response method.
There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer strategic-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. We present an approach for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically-constrained program based on a nonlinear complementarity problem formulation from the sequence-form game representation. This approach capitalizes on recent advances for solving nonconvex quadratic programs. Our algorithm is able to quickly solve three-player Kuhn poker after removal of dominated actions. Of the available algorithms in the Gambit software suite, only the logit quantal response approach is successfully able to solve the game; however, the approach takes longer than our algorithm and also involves a degree of approximation. Our formulation also leads to a new approach for computing Nash equilibrium in multiplayer strategic-form games which we demonstrate to outperform a previous quadratically-constrained program formulation.