# Computing Bayes-Nash Equilibria in Combinatorial Auctions with Verification

## TL;DR

This paper introduces a fast, verified algorithm for computing approximate Bayes-Nash equilibria in complex combinatorial auctions with continuous values, enabling analysis of more intricate auction domains.

## Contribution

The paper's main novelty is a separation of search and verification phases in computing equilibria, with a new verification method providing theoretical guarantees without mechanism assumptions.

## Key findings

- Efficient algorithm for $ ext{ε}$-BNE computation in complex auctions.
- Theoretical bounds on approximation error across continuous value spaces.
- Open-source implementation facilitating research and strategy analysis.

## Abstract

We present a new algorithm for computing pure-strategy $\varepsilon$-Bayes-Nash equilibria ($\varepsilon$-BNEs) in combinatorial auctions with continuous value and action spaces. An essential innovation of our algorithm is to separate the algorithm's search phase (for finding the $\varepsilon$-BNE) from the verification phase (for computing the $\varepsilon$). Using this approach, we obtain an algorithm that is both very fast and provides theoretical guarantees on the $\varepsilon$ it finds. Our main technical contribution is a verification method which allows us to upper bound the $\varepsilon$ across the whole continuous value space without making assumptions about the mechanism. Using our algorithm, we can now compute $\varepsilon$-BNEs in multi-minded domains that are significantly more complex than what was previously possible to solve. We release our code under an open-source license to enable researchers to perform algorithmic analyses of auctions, to enable bidders to analyze different strategies, and to facilitate many other applications.