We define and characterize a notion of correlated equilibrium for games with incomplete information, which we call Bayes correlated equilibrium: The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may have access to additional signals beyond the information structure is characterized and shown to be equivalent to the set of Bayes correlated equilibria.
A game of incomplete information can be decomposed into a basic game, given by actions sets and payoff functions, and an information structure. We introduce a partial order on many player information structures — which we call individual sufficiency — under which more information shrinks the set of Bayes correlated equilibria. We discuss the relation of the solution concept to alternative definitions of correlated equilibrium in incomplete information games and of the partial order on information structures to others, including Blackwell’s for the single player case.
The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may have access to additional signals beyond the given information structure is equivalent to the set of a version of incomplete information correlated equilibrium which we dub Bayes correlated equilibrium.
A game of incomplete information can be decomposed into a basic game, given by actions sets and payoff functions, and an information structure. We identify a partial order on many player information structures (individual sufficiency) under which more information shrinks the set of Bayes correlated equilibria.
The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may have access to additional signals beyond the given information structure is characterized and shown to be equivalent to the set of a version of incomplete information correlated equilibria which we dub Bayes correlated equilibria.
A game of incomplete information can be decomposed into a basic game, given by actions sets and payoff functions, and an information structure. We introduce a partial order on many player information structures — which we call individual sufficiency — under which more information shrinks the set of Bayes correlated equilibria.
We discuss the relation of the solution concept to alternative definitions of correlated equilibrium in incomplete information games and of the partial order on information structures to others, including Blackwell’s for the single player case.
A game of incomplete information can be decomposed into a basic game and an information structure. The basic game defines the set of actions, the set of payoff states the payoff functions and the common prior over the payoff states. The information structure refers to the signals that the players receive in the game.
We characterize the set of outcomes that can arise in Bayes Nash equilibrium if players observe the given information structure but may also observe additional signals. The characterization corresponds to the set of (a version of) incomplete information correlated equilibria which we dub Bayes correlated equilibria.
We identify a partial order on many player information structures (individual sufficiency) under which more information shrinks the set of Bayes correlated equilibria. This order captures the role of information in imposing (incentive) constraints on behavior.
We consider the design of an optimal auction in which the seller can determine the allocation and the disclosure rule of the mechanism. Thus, in contrast to the standard analysis of a optimal auctions, the seller can explicitly design the disclosure of the information received by each bidder as his private information.
We show that the optimal disclosure rule is a sequential disclosure rule, implemented in an ascending price auction. In the optimal disclosure mechanism, each losing bidder learns his true valuation, but the winning bidder only learns that his valuation is sufficiently high to win the auction. We show that in the optimal auction, the posterior incentive and participation constraints of all the bidders are satisfied. In the special case in which the bidders have no private information initially, the seller can extract the entire surplus.
We propose a sequential auction mechanism for a single object in which the seller jointly determines the allocation and the disclosure policy. A sequential disclosure rule is shown to implement an ascending price auction in which each losing bidder learns his true valuation, but the winning bidder’s information is truncated from below. As the auction ends, the winning bidder only has limited information, namely that his valuation is sufficiently high to win the auction. The sequential mechanism implements the allocation of the handicap auction of Esö and Szentes [10] but strengthens the participation constraints of the bidders from interim to posterior constraints. Due to the limited disclosure of information, the participation constraints (and incentive constraints) of all the bidders are satisfied with respect to all information revealed by the mechanism. In the special case in which the bidders have no private information initially, the seller can extract the entire surplus.
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.”
We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the efficient gains from trade.
As well as characterizing the welfare impact of price discrimination, we examine the limits of how prices and quantities can change under price discrimination. We also examine the limits of price discrimination in richer environments with quantity discrimination and limited ability to segment the market.
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.”
We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the surplus generated by efficient trade.
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.”
We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the efficient gains from trade.
As well as characterizing the welfare impact of price discrimination, we examine the limits of how prices and quantities can change under price discrimination. We also examine the limits of price discrimination in richer environments with quantity discrimination and limited ability to segment the market.
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge different prices to different segments of the market, i.e., carry out “third degree price discrimination.” We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as profits under the uniform monopoly price, and (iii) total surplus does not exceed the surplus generated by efficient trade.
We analyze a nonlinear pricing model with limited information. Each buyer can purchase a large variety, d, of goods. His preference for each good is represented by a scalar and his preference over d goods is represented by a d-dimensional vector. The type space of each buyer is given by a compact subset of Rd+ with a continuum of possible types. By contrast, the seller is limited to offer a finite number M of d-dimensional choices.
We provide necessary conditions that the optimal finite menu of the social welfare maximizing problem has to satisfy. We establish an underlying connection to the theory of quantization and provide an estimate of the welfare loss resulting from the usage of the d-dimensional M-class menu. We show that the welfare loss converges to zero at a rate proportional to d/M2/d.
We show that in higher dimensions, a significant reduction in the welfare loss arises from an optimal partition of the d-dimensional type space that takes advantage of the correlation among the d parameters.
(with Stephen Morris) “Equilibrium robustness in informational variables is critical, if one wants to use results from the mechanism design literature in real life applications. The papers included in the Bergemann and Morris book describe state of the art progress in this direction of research. The book is an excellent resource for established game theorists, who want to learn more about this subject; and for PhD students, who look for exciting problems to investigate.” ─ Ehud Kalai, Kellogg School of Management, Northwestern University
“This book collects together a series of papers on mechanism design written by Dirk Bergemann and Stephen Morris. It is their response to the challenge set by Robert Wilson in his eponymous doctrine: Only by repeated weakening of common knowledge assumptions will the theory approximate reality. Many scholars responded by arguing for solution concepts robust to the beliefs of the agents. The approach taken by Bergemann and Morris was radically different. They hitched their wagon to Harsany’s observation that relaxing the common knowledge assumption was equivalent to enlarging the type space. Then, they proceed to develop the properties of mechanisms that would emerge. For this reason, this collection is essential reading for any student interested in taking up the challenge of the Wilson doctrine. The introduction by itself is worth the price of admission!” ─ Rakesh Vohra, Kellogg School of Management, Northwestern University
“Mechanism design has been one of the great successes of economic theory in the last 30 years. Robust mechanism design, the study of optimal mechanisms in settings where the designer has less information about the beliefs of the agents, is the natural next step in the evolution of this field. Bergemann and Morris are two of the leading figures in developing this new theory, and this book combines many of their papers with an excellent introduction that overviews the field and explains how their papers fit together. Highly recommended to all students and practitioners of economic theory, and essential reading for would-be mechanism designers.” ─ Drew Fudenberg, Department of Economics, Harvard University
“The question of the design of institutions has been at the center of some of the most important economic theory in the past four decades. Bergemann and Morris have made seminal contributions to the understanding of how uncertainty can and should be incorporated into mechanism design, and this volume reproduces a collection of their most important work in the area. The volume will be an important reference for those working in the area and those who wish to apply the ideas in economic models.” ─ Andrew Postlewaite, Department of Economics, University of Pennsylvania