Computing Bayes-Nash Equilibria in Combinatorial Auctions with Verification

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.