Computing Bayes Nash Equilibrium Strategies in Auction Games via Simultaneous Online Dual Averaging

Auctions are modeled as Bayesian games with continuous type and action spaces. Computing equilibria in auction games is computationally hard in general and no exact solution theory is known. We introduce algorithms computing distributional strategies on a discretized version of the game via online convex optimization. One advantage of distributional strategies is that we do not have to make any assumptions on the shape of the bid function. Besides, the expected utility of agents is linear in the strategies. It follows that if our regularized optimization algorithms converge to a pure strategy, then they converge to an approximate equilibrium of the discretized game with high precision. Importantly, we show that the equilibrium of the discretized game approximates an equilibrium in the continuous game. In a wide variety of auction games, we provide empirical evidence that the method approximates the analytical (pure) Bayes Nash equilibrium closely. This speed and precision is remarkable, because in many finite games learning dynamics do not converge or are even chaotic. In standard models where agents are symmetric, we find equilibrium in seconds. The method allows for interdependent valuations and different types of utility functions and provides a foundation for broadly applicable equilibrium solvers that can push the boundaries of equilibrium analysis in auction markets and beyond.