Computing Perfect Bayesian Equilibria in Sequential Auctions

We present a best-response based algorithm for computing verifiable $\varepsilon$-perfect Bayesian equilibria for sequential auctions with combinatorial bidding spaces and incomplete information. Previous work has focused only on computing Bayes-Nash equilibria for static single-round auctions, which our work captures as a special case. Additionally, we prove an upper bound $\varepsilon$ on the utility loss of our approximate equilibria and present an algorithm to efficiently compute $\varepsilon$ based on the immediate loss at each subgame. We evaluate the performance of our algorithm by reproducing known results from several auctions previously introduced in the literature, including a model of combinatorial split-award auctions used in procurement.Comment: 12 pages, 8 figure

Computing Bayes Nash Equilibrium Strategies in Auction Games via Simultaneous Online Dual Averaging

Auctions are modeled as Bayesian games with continuous type and action spaces. Computing equilibria in auction games is computationally hard in general and no exact solution theory is known. We introduce algorithms computing distributional strategies on a discretized version of the game via online convex optimization. One advantage of distributional strategies is that we do not have to make any assumptions on the shape of the bid function. Besides, the expected utility of agents is linear in the strategies. It follows that if our regularized optimization algorithms converge to a pure strategy, then they converge to an approximate equilibrium of the discretized game with high precision. Importantly, we show that the equilibrium of the discretized game approximates an equilibrium in the continuous game. In a wide variety of auction games, we provide empirical evidence that the method approximates the analytical (pure) Bayes Nash equilibrium closely. This speed and precision is remarkable, because in many finite games learning dynamics do not converge or are even chaotic. In standard models where agents are symmetric, we find equilibrium in seconds. The method allows for interdependent valuations and different types of utility functions and provides a foundation for broadly applicable equilibrium solvers that can push the boundaries of equilibrium analysis in auction markets and beyond

On the beliefs off the path: equilibrium refinement due to quantal response and level-k

This paper studies the relevance of equilibrium and nonequilibrium explanations of behavior, with respects to equilibrium refinement, as players gain experience. We investigate this experimentally using an incomplete information sequential move game with heterogeneous preferences and multiple perfect equilibria. Only the limit point of quantal response (the limiting logit equilibrium), and alternatively that of level-k reasoning (extensive form rationalizability), restricts beliefs off the equilibrium path. Both concepts converge to the same unique equilibrium, but the predictions differ prior to convergence. We show that with experience of repeated play in relatively constant environments, subjects approach equilibrium via the quantal response learning path. With experience spanning also across relatively novel environments, though, level-k reasoning tends to dominate

Sequential Two-Player Games with Ambiguity

If players' beliefs are strictly non-additive, the Dempster-Shafer updating rule can be used to define beliefs off the equilibrium path. We define an equilibrium concept in sequential two-person games where players update their beliefs with the Dempster-Shafer updating rule. We show that in the limit as uncertainty tends to zero, our equilibrium approximates Bayesian Nash equilibrium by imposing context-dependent constraints on beliefs under uncertainty.