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

arXiv: https://arxiv.org/abs/2208.02036 · HTML: https://mlsanity.com/paper/2208.02036/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Aug 2022.

## Built on by (most cited first, 6 of 6)

- [Joint Value Estimation and Bidding in Repeated First-Price Auctions](https://mlsanity.com/paper/2502.17292/) (2025)
- [Learning to Bid in Non-Stationary Repeated First-Price Auctions](https://mlsanity.com/paper/2501.13358/) (2025)
- [Understanding Iterative Combinatorial Auction Designs via Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2402.19420/) (2024)
- [Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks](https://mlsanity.com/paper/2211.15936/) (2022)
- [ApproxED: Approximate exploitability descent via learned best responses](https://mlsanity.com/paper/2301.08830/) (2023)
- [Safe Exploitative Play with Untrusted Type Beliefs](https://mlsanity.com/paper/2411.07679/) (2024)

## Abstract (arXiv)

Auctions are modeled as Bayesian games with continuous type and action spaces. Determining equilibria in auction games is computationally hard in general and no exact solution theory is known. We introduce an algorithmic framework in which we discretize type and action space and then learn distributional strategies via online optimization algorithms. 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 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 approach 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. While we focus on dual averaging, we show that the overall approach converges independent of the regularizer and alternative online convex optimization methods achieve similar results, even though the discretized game neither satisfies monotonicity nor variational stability globally. 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.