Computing Bayes-Nash Equilibria in Combinatorial Auctions with Verification

We present a new algorithm for computing pure-strategy ε-Bayes-Nash equilibria (ε-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 ε-BNE) from the verification phase (for computing the ε). Using this approach, we obtain an algorithm that is both very fast and provides theoretical guarantees on the ε it finds. Our main technical contribution is a verification method which allows us to upper bound the ε across the whole continuous value space without making assumptions about the mechanism. Using our algorithm, we can now compute ε-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.