Hu Fu

Zhicheng Du, Hu Fu, Ying Qin et al.

Advertisements often strategically disclose information to consumers who make decisions on further information acquisition and eventual purchase. Anderson and Renault (2006) model this problem using an information design framework, where the advertiser acts as a sender and the consumer as a receiver. We extend this model to a competitive setting with horizontally differentiated senders competing for a unit-demand receiver. Under costly inspection, the receiver's optimal sequential search action is given by Weitzman's Index Algorithm. We give a method, based on duality arguments, to verify whether a sender's given information strategy constitutes a best response against his competitors (other senders). We establish the existence of an equilibrium in the game among senders when the prior distributions have no mass; we also illustrate that such equilibria may exhibit intricate behaviors. Finally, we meticulously characterize symmetric equilibria played by the senders for cases when the prior distributions have monotone increasing densities, while offering economic intuitions behind the insightful equilibrium structure.

Hu Fu, Tao Lin

In non-truthful auctions such as first-price and all-pay auctions, the independent strategic behaviors of bidders, with the corresponding Bayes-Nash equilibrium notion, are notoriously difficult to characterize and can cause undesirable outcomes. An alternative approach to achieve better outcomes in non-truthful auctions is to coordinate the bidders: let a mediator make incentive-compatible recommendations of correlated bidding strategies to the bidders, namely, implementing a Bayes correlated equilibrium (BCE). The implementation of BCE, however, requires knowledge of the distributions of bidders' private valuations, which is often unavailable. We initiate the study of the sample complexity of learning Bayes correlated equilibria in non-truthful auctions. We prove that the set of strategic-form BCEs in a large class of non-truthful auctions, including first-price and all-pay auctions, can be learned with a polynomial number $\tilde O(\frac{n}{\varepsilon^2})$ of samples of bidders' values. This moderate number of samples demonstrates the statistical feasibility of learning to coordinate bidders. Our technique is a reduction to the problem of estimating bidders' expected utility from samples, combined with an analysis of the pseudo-dimension of the class of all monotone bidding strategies.

Amit Kadan, Hu Fu

Motivated by Generative Adversarial Networks, we study the computation of Nash equilibrium in concave network zero-sum games (NZSGs), a multiplayer generalization of two-player zero-sum games first proposed with linear payoffs. Extending previous results, we show that various game theoretic properties of convex-concave two-player zero-sum games are preserved in this generalization. We then generalize last iterate convergence results obtained previously in two-player zero-sum games. We analyze convergence rates when players update their strategies using Gradient Ascent, and its variant, Optimistic Gradient Ascent, showing last iterate convergence in three settings -- when the payoffs of players are linear, strongly concave and Lipschitz, and strongly concave and smooth. We provide experimental results that support these theoretical findings.

Hu Fu, Tao Lin

In non-truthful auctions, agents' utility for a strategy depends on the strategies of the opponents and also the prior distribution over their private types; the set of Bayes Nash equilibria generally has an intricate dependence on the prior. Using the First Price Auction as our main demonstrating example, we show that $\tilde O(n / ε^2)$ samples from the prior with $n$ agents suffice for an algorithm to learn the interim utilities for all monotone bidding strategies. As a consequence, this number of samples suffice for learning all approximate equilibria. We give almost matching (up to polylog factors) lower bound on the sample complexity for learning utilities. We also consider a setting where agents must pay a search cost to discover their own types. Drawing on a connection between this setting and the first price auction, discovered recently by Kleinberg et al. (2016), we show that $\tilde O(n / ε^2)$ samples suffice for utilities and equilibria to be estimated in a near welfare-optimal descending auction in this setting. En route, we improve the sample complexity bound, recently obtained by Guo et al. (2021), for the Pandora's Box problem, which is a classical model for sequential consumer search.