Learning equilibrium in bilateral bargaining games

Bilateral bargaining of a single good among one buyer and one seller describes the simplest form of trade, yet Bayes–Nash equilibrium strategies are largely unknown. Only for the average mechanism in the standard independent private values model with independent and uniform priors, we know that there is a continuum of equilibria. However, a non-uniform prior distribution already leads to a system of non-linear differential equations for which closed-form bidding strategies cannot be derived. Recent advances in equilibrium learning provide a numerical approach to equilibrium analysis, which can push the boundaries of existing results and allow for the analysis of environments that have been considered intractable so far. We study Neural Pseudogradient Ascent (NPGA) and Simultaneous Online Dual Averaging (SODA), two new equilibrium learning algorithms for Bayesian auction games with continuous type and action spaces. Although the environment is simple to describe, the continuum of equilibria makes it challenging for equilibrium learning algorithms. Empirically, NPGA finds the payoff-maximizing linear equilibrium, while SODA also finds non-differentiable step-function equilibria. Interestingly, the algorithms also find equilibrium with non-uniform priors and risk-averse traders for which we do not know an analytical solution. We show that the game is not globally monotone, but we can prove local convergence for a model with uniform priors and linear bid functions.