Uniqueness of the Equilibrium in First-Price Auctions

If the value cumulative distribution functions are log-concave at the highest lower extremity of their supports of the first-price auction in the asymmetric indepent private values model.

Uniqueness of the Equilibrium in First-Price Auctions

If the value cumulative distribution functions are log-concave at the highest lower extremity of their supports, a simple geometric argument establishes the uniqueness of the equilibrium of the first-price auction in the asymmetric independent private value model.

Multiple equilibria in asymmetric first-price auctions

Maskin and Riley (2003) and Lebrun (2006) prove that the Bayes-Nash equilibrium of �rst-price auctions is unique. This uniqueness requires the assumption that a buyer never bids above his value. We demonstrate that, in asymmetric �rst-price auctions (with or without a minimum bid), the relaxation of this assumption results in additional equilibria that are "substantial." Although in each of these additional equilibria no buyer wins with a bids above his value, the allocation of the object and the selling price may vary among the equilibria. Furthermore, we show that such phenomena can only occur under asymmetry in the distributions of values.Asymmetric auctions, �first-price auctions, multiple equilibria

First price auctions: Monotonicity and uniqueness

I study monotonicity and uniqueness of the equilibrium strategies in a two-person first price auction with affiliated signals. I show that when the game is symmetric there is a unique Nash equilibrium that satisfies a regularity condition requiring that the equilibrium strategies be {\sl piecewise monotone}. Moreover, when the signals are discrete-valued, the equilibrium is unique. The central part of the proof consists of showing that at any regular equilibrium the bidders' strategies must be monotone increasing within the support of winning bids. The monotonicity result derived in this paper provides the missing link for the analysis of uniqueness in two-person first price auctions. Importantly, this result extends to asymmetric auctions.Auctions, game theory

Pacing Equilibrium in First-Price Auction Markets

In the isolated auction of a single item, second price often dominates first price in properties of theoretical interest. But, single items are rarely sold in true isolation, so considering the broader context is critical when adopting a pricing strategy. In this paper, we study a model centrally relevant to Internet advertising and show that when items (ad impressions) are individually auctioned within the context of a larger system that is managing budgets, theory offers surprising endorsement for using a first price auction to sell each individual item. In particular, first price auctions offer theoretical guarantees of equilibrium uniqueness, monotonicity, and other desirable properties, as well as efficient computability as the solution to the well-studied Eisenberg-Gale convex program. We also use simulations to demonstrate that a bidder's incentive to deviate vanishes in thick markets

Thresholding at the monopoly price: an agnostic way to improve bidding strategies in revenue-maximizing auctions

We address the problem of improving bidders' strategies in prior-dependent revenue-maximizing auctions. We introduce a simple and generic method to design novel bidding strategies if the seller uses past bids to optimize her mechanism. This strategy works with general value distributions, with asymmetric bidders and for different revenue-maximizing mechanisms. Furthermore, it can be made robust to sample approximation errors on the seller part. This results in a large increase in utility for bidders whether they have a full or partial knowledge of their competitors. In the case where the buyer has no information about the competition, we propose a simple and agnostic strategy that is robust to mechanism changes and local (as opposed to global) optimization of e.g. reserve prices by the seller. In textbook-style examples, for instance with uniform value distributions and two bidders, this no-side-information and mechanism-independent strategy yields an enormous 57% increase in buyer utility for lazy second price auctions with no reserves. In the i.i.d symmetric case, we show existence and uniqueness of a Nash equilibrium in the class of strategy we consider for lazy second price auctions, as well as the corresponding explicit shading strategies. Our approach also works for Myerson auctions for instance. At this Nash equilibrium, buyer's utility is the same as in a second price auction with no reserve. Our approach also yields optimal solutions when buyer are constrained in the class of shading strategies they can use, a realistic constraint in practical applications. The heart of our approach is to see optimal auctions in practice as a Stackelberg game where the buyer is the leader, as he is the first one to move (here bid) when the seller is the follower as she has no prior information on the bidder

Price Competition in Online Combinatorial Markets

We consider a single buyer with a combinatorial preference that would like to purchase related products and services from different vendors, where each vendor supplies exactly one product. We study the general case where subsets of products can be substitutes as well as complementary and analyze the game that is induced on the vendors, where a vendor's strategy is the price that he asks for his product. This model generalizes both Bertrand competition (where vendors are perfect substitutes) and Nash bargaining (where they are perfect complements), and captures a wide variety of scenarios that can appear in complex crowd sourcing or in automatic pricing of related products. We study the equilibria of such games and show that a pure efficient equilibrium always exists. In the case of submodular buyer preferences we fully characterize the set of pure Nash equilibria, essentially showing uniqueness. For the even more restricted "substitutes" buyer preferences we also prove uniqueness over {\em mixed} equilibria. Finally we begin the exploration of natural generalizations of our setting such as when services have costs, when there are multiple buyers or uncertainty about the the buyer's valuation, and when a single vendor supplies multiple products.Comment: accept to WWW'14 (23rd International World Wide Web Conference