We present an algorithm for computing pure-strategy epsilon-perfect Bayesian equilibria in sequential auctions with continuous action and value spaces. Importantly, our algorithm includes a verification phase that computes an upper bound on the utility loss of the found strategies. Prior work on equilibrium computation in auctions with verification has focussed on the single-round case, but the methods do not work for sequential auctions because of two main challenges: (1) there are infinitely many subgames, and (2) the setting has no optimal substructure as bidders' beliefs and best response strategies depend on the strategies of previous rounds. We make two contributions. First, we introduce a tailor-made game abstraction that discretizes the auction and augments the state space with the public beliefs, such that an approximate equilibrium can be computed via dynamic programming. Second, we prove a decomposition theorem to upper bound the utility loss of the computed equilibrium. This is essential because it is neither guaranteed that the auction has an equilibrium nor that any algorithm converges to it. We validate our algorithm on multiple settings with known equilibria and apply it to a new multi-round combinatorial auction.

We study a class of manipulations in combinatorial auctions where bidders fundamentally misrepresent what goods they are interested in. Prior work has largely assumed that bidders only submit bids on their bundles of interest, which we call simple bidding: strategizing over the bid amounts, but not the bundle identities. However, we show that there exists an entire class of auction instances for which simple bids are never optimal in BNE, always being strictly dominated by complex bids (where bidders bid on goods they are not interested in). We show this result for the two most widely used auction mechanisms:first price andVCG-nearest. We also explore the structural properties of the winner determination problem that cause this phenomenon, and we use the insights gained to investigate how impactful complex bidding manipulations may be. We find that, in the worst case, a bidder's optimal complex bid may require bidding on an exponential number of bundles, even if the bidder is interested only in a single good. Thus, this phenomenon can greatly impact the auction's outcome, and should not be ignored by bidders and auction designers alike.

We present a new algorithm for computing pure-strategy ε-Bayes-Nash equilibria (ε-BNEs) in combinatorial auctions. The main 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 contribution is a verification method which, surprisingly, 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 many other applications.

Combinatorial auctions are used to allocate resources in domains where bidders have complex preferences over bundles of goods. However, the behavior of bidders under different payment rules is not well understood, and there has been limited success in finding Bayes-Nash equilibria of such auctions due to the computational difficulties involved. In this paper, we introduce non-decreasing payment rules. Under such a rule, the payment of a bidder cannot decrease when he increases his bid, which is a natural and desirable property. VCG-nearest, the payment rule most commonly used in practice, violates this property and can thus be manipulated in surprising ways. In contrast, we show that many other payment rules are non-decreasing. We also show that a non-decreasing payment rule imposes a structure on the auction game that enables us to search for an approximate Bayes-Nash equilibrium much more efficiently than in the general case. Finally, we introduce the utility planes BNE algorithm, which exploits this structure and outperforms a state-of-the-art algorithm by multiple orders of magnitude.

Combinatorial auctions (CAs) are widely used in practice, which is why understanding their incentive properties is an important problem. However, finding Bayes-Nash equilibria (BNEs) of CAs analytically is tedious, and prior algorithmic work has only considered limited solution concepts (e.g. restricted action spaces). In this paper, we present a fast, general algorithm for computing symmetric pure ε-BNEs in CAs with continuous values and actions. In contrast to prior work, we separate the search phase (for finding the BNE) from the verification step (for estimating the ε), and always consider the full (continuous) action space in the best response computation. We evaluate our method in the well-studied LLG domain, against a benchmark of 16 CAs for which analytical BNEs are known. In all cases, our algorithm converges quickly, matching the known results with high precision. Furthermore, for CAs with quasi-linear utility functions and independently distributed valuations, we derive a theoretical bound on ε. Finally, we introduce the new Multi-Minded LLLLGG domain with eight goods and six bidders, and apply our algorithm to finding an equilibrium in this domain. Our algorithm is the first to find an accurate BNE in a CA of this size.