A Fischer–Burmeister Merit Function Approach to Neural Network Approximation of Bayesian Nash Equilibria in Bayesian Games with Continuous Actions and Types

Bayesian games model interactions where players possess private information and have incomplete information about each other. Computing Bayesian Nash equilibria (BNE) in continuous Bayesian games remains challenging due to infinite-dimensional strategy spaces, multi-objective conditions, and intractable expected utilities. To address these computational bottlenecks, this paper proposes a novel learning framework. First, we reformulate the game’s equilibrium problem into a tractable single-objective minimization task by leveraging the Fischer–Burmeister function to combine all players’ Karush–Kuhn–Tucker (KKT) conditions into a global merit function. Second, we introduce a dual neural network architecture per player—coupling strategy and Lagrange networks—to effectively approximate continuous strategy spaces. Third, we integrate Monte Carlo sampling to tractably evaluate expected utilities during training. We theoretically establish the asymptotic convergence of the cost function under stochastic gradient descent (SGD) and show that, conditional on the trained networks achieving a sufficiently small empirical merit loss, the resulting strategy profiles constitute an ϵ-Bayesian Nash equilibrium (ϵ-BNE) under sufficiently large sample sizes. Experiments and theoretical analysis demonstrate the effectiveness of the proposed approach for continuous incomplete-information games with strongly concave utilities that are thrice continuously differentiable and have bounded first and second derivatives.