Spatial Branch-and-Bound for Computing Multiplayer Nash Equilibrium
Jakub Černý, Shuvomoy Das Gupta, Christian Kroer
TL;DR
This work targets the challenge of computing Nash equilibria in multiplayer normal-form games, where general-sum settings are $PPAD$-complete and complete solvers struggle to scale. The authors formulate NE as a polynomial complementarity problem and develop a complete, sound spatial branch-and-bound (SBnB) algorithm, leveraging a penalized continuous reformulation to enable efficient continuous-branching and strong warm-starts from nonlinear interior-point methods. They prove equivalence between the continuous penalized formulation and the original MIQCP, and show that a zero penalization corresponds to an exact NE, while small penalties yield $\epsilon$-Nash equilibria. Empirically, the proposed SBnB (and its early-termination variant SBnB-e) outperforms existing complete methods on mid-sized random graphical games, delivering higher-precision equilibria and faster runtimes, and demonstrating the potential for integration into oracle-based solvers for larger-scale problems.
Abstract
Equilibria of realistic multiplayer games constitute a key solution concept both in practical applications, such as online advertising auctions and electricity markets, and in analytical frameworks used to study strategic voting in elections or assess policy impacts in integrated assessment models. However, efficiently computing these equilibria requires games to have a carefully designed structure and satisfy numerous restrictions; otherwise, the computational complexity becomes prohibitive. In particular, finding even approximate Nash equilibria in general-sum normal-form games with two or more players is known to be PPAD-complete. Current state-of-the-art algorithms for computing Nash equilibria in multiplayer normal-form games either suffer from poor scalability due to their reliance on non-convex optimization solvers, or lack guarantees of convergence to a true equilibrium. In this paper, we propose a formulation of the Nash equilibrium computation problem as a polynomial complementarity problem and develop a complete and sound spatial branch-and-bound algorithm based on this formulation. We provide a qualitative analysis arguing why one should expect our approach to perform well, and show the relationship between approximate solutions to our formulation and that of computing an approximate Nash equilibrium. Empirical evaluations demonstrate that our algorithm substantially outperforms existing complete methods.