# Bayesian Nash Equilibrium in First-Price Auction with Discrete Value Distributions

arXiv: https://arxiv.org/abs/1906.09403 · HTML: https://mlsanity.com/paper/1906.09403/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2019.

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

- [Learning to Bid in Contextual First Price Auctions](https://mlsanity.com/paper/2109.03173/) (2021)
- [Nash Convergence of Mean-Based Learning Algorithms in First-Price Auctions](https://mlsanity.com/paper/2110.03906/) (2021)
- [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)
- [Optimizing Bidding Strategies in First-Price Auctions in Binary Feedback Setting with Predictions](https://mlsanity.com/paper/2506.15817/) (2025)
- [Learning to Coordinate Bidders in Non-Truthful Auctions](https://mlsanity.com/paper/2507.02801/) (2025)

## Abstract (arXiv)

First price auctions are widely used in government contracts and industrial auctions. In this paper, we consider the Bayesian Nash Equilibrium (BNE) in first price auctions with discrete value distributions. We study the characterization of the BNE in the first price auction and provide an algorithm to compute the BNE at the same time. Moreover, we prove the existence and the uniqueness of the BNE. Some of the previous results in the case of continuous value distributions do not apply to the case of discrete value distributions. In the meanwhile, the uniqueness result in discrete case cannot be implied by the uniqueness property in the continuous case. Unlike in the continuous case, we do not need to solve ordinary differential equations and thus do not suffer from the solution errors therein. Compared to the method of using continuous distributions to approximate discrete ones, our experiments show that our algorithm is both faster and more accurate. The results in this paper are derived in the asymmetric independent private values model, which assumes that the buyers' value distributions are common knowledge.