Crowdsourced Bayesian auctions

We investigate the problem of optimal mechanism design, where an auctioneer wants to sell a set of goods to buyers, in order to maximize revenue. In a Bayesian setting the buyers' valuations for the goods are drawn from a prior distribution D, which is often assumed to be known by the seller. In this work, we focus on cases where the seller has no knowledge at all, and "the buyers know each other better than the seller knows them". In our model, D is not necessarily common knowledge. Instead, each buyer individually knows a posterior distribution associated with D. Since the seller relies on the buyers' knowledge to help him set a price, we call these types of auctions crowdsourced Bayesian auctions. For this crowdsourced Bayesian model and many environments of interest, we show that, for arbitrary valuation distributions D (in particular, correlated ones), it is possible to design mechanisms matching to a significant extent the performance of the optimal dominant-strategy-truthful mechanisms where the seller knows D. To obtain our results, we use two techniques: (1) proper scoring rules to elicit information from the players; and (2) a reverse version of the classical Bulow-Klemperer inequality. The first lets us build mechanisms with a unique equilibrium and good revenue guarantees, even when the players' second and higher-order beliefs about each other are wrong. The second allows us to upper bound the revenue of an optimal mechanism with n players by an n/n--1 fraction of the revenue of the optimal mechanism with n -- 1 players. We believe that both techniques are new to Bayesian optimal auctions and of independent interest for future work.United States. Office of Naval Research (Grant number N00014-09-1-0597

Robust Mechanism Design: An Introduction

This essay is the introduction for a collection of papers by the two of us on "Robust Mechanism Design" to be published by World Scientific Publishing. The appendix of this essay lists the chapters of the book. The objective of this introductory essay is to provide the reader with an overview of the research agenda pursued in the collected papers. The introduction selectively presents the main results of the papers, and attempts to illustrate many of them in terms of a common and canonical example, the single unit auction with interdependent values. In addition, we include an extended discussion about the role of alternative assumptions about type spaces in our work and the literature, in order to explain the common logic of the informational robustness approach that unifies the work in this volume.Mechanism design, Robust mechanism design, Common knowledge, Universal type space, Interim equilibrium, Ex post equilibrium, Dominant strategies, Rationalizability, Partial implementation, Full implementation, Robust implementation

Robust Mechanism Design: An Introduction

This essay is the introduction for a collection of papers by the two of us on “Robust Mechanism Design” to be published by World Scientific Publishing. The appendix of this essay lists the chapters of the book. The objective of this introductory essay is to provide the reader with an overview of the research agenda pursued in the collected papers. The introduction selectively presents the main results of the papers, and attempts to illustrate many of them in terms of a common and canonical example, the single unit auction with interdependent values. In addition, we include an extended discussion about the role of alternative assumptions about type spaces in our work and the literature, in order to explain the common logic of the informational robustness approach that unifies the work in this volume

Information uncertainty in auction theory

van Welbergen N. Information uncertainty in auction theory. Bielefeld: Universität Bielefeld; 2016.For a long time in the literature, the usual way of modeling an auction participant's beliefs has been exogenous - the distribution of private value is simply given, fixed, commonly known and out of any doubt for the involved participants. Growing awareness on information uncertainty and its strong influence on decision choices and social outcomes is the reason why we focus on introducing information uncertainty in auction theory. In particular, we allow involved subjects to question underlying information structure. We propose three different ways to design the endogenous belief formation process and explore its influence on state of the art in auction theory. This research approach results in three different setups, each examined in depth in a separate chapter. Chapter 2 shows the importance of our modeling approach. The genericity of an optimal auction format - in the literature known as the Crémer-McLean auction - is challenged and its robustness against a change of information structure is examined. We show that participation constraints in the auction format fail to hold once the seller and bidders have slightly different beliefs over the joint distribution of private values. We propose the definition of a belief neighborhood in order to capture this slight discrepancy between seller's and bidders' beliefs. Moreover, we also give a quantitative assessment of how often the failure occurs in the case of a common prior assumption as well as once the common prior assumption among bidders is relaxed. In particular, participation constraints fail in at least one half of any plausible belief's neighborhood. Once the common prior assumption among bidders is relaxed, the failure occurs everywhere except in 1/(2^(2m)) of a belief's neighborhood, where m is the number of possible private values for each bidder. The origin of private values in auction models is examined in Chapter 3. In contrast with the standard literature, we let participants question the information structure involved in auctions. We focus on the most prevalent forms of sealed-bid auctions on the market: the first-price and second-price auctions. Instead of being exogenously given, we permit the observed distribution of private value to take the form of mixture probability distribution, which in turn allows participants to speculate on the true distribution of the values. The speculations are done in accordance with Bayes' rule - the bidders use their private value to infer the distribution, whereas the seller bases his belief on an observed mixture form of the values' distribution. We explore the consequences of allowing this particular belief formation on bidders' and seller's behavior. First of all, despite the independently and identically distributed values, we show that the belief formation leads to the revenue equivalence failure in such a way that the seller prefers the second-price auction. Moreover, in our setup, truthfully bidding one's own private value continues to be optimal in the second-price auction. However, the bidding behavior in the first-price auction is influenced by the belief formation in the following way. Comparison of bidding strategies in the standard framework and our model leads to the statement that these two strategies cross at most once. In other words, we discover that there is either dominance ordering or a single-crossing property between the compared strategies. The final chapter presents a model with a new kind of information uncertainty - the uncertainty about seller's type. Motivated by recent security and fraud issues in auctions, we develop a model where a manipulative seller has an opportunity to send an agent to bid secretly on the seller's behalf. Out of the possible auction formats, we choose the second-price auction and the all-pay auction. We look at the all-pay auction because of its design: everyone pays her own bid, irrespective of whether she wins the auction. Consequently, it seems that the manipulative seller can make the best use of his manipulation in the all-pay auction. In the all-pay auction, the seller may keep the object and collect all bids, which is not the case with either a second-price or first-price auction. In this way we change the standard information structure of an auction by proposing that the seller also holds private information, whether or not he is a manipulative (with an agent) or an honest seller (without an agent). Even though a priori there is a common prior over the possibility of facing a cheating seller, bidders in our model perceive the choice of auction format as a signal about the seller's intentions. Thus, our setup extends the standard auction framework to a form of signaling game. To this end, we explore the pure weak perfect Bayesian Nash equilibria of this game. We show that the only robust weak Bayesian Nash equilibrium is pooling on the second-price: that is, it is beneficial for both type of sellers to choose the second-price auction. In addition, there is a special non-generic equilibrium scenario in which the honest seller chooses the all-pay auction and the cheating seller chooses the second-price auction. However, this equilibrium is very unstable and not robust against the smallest change of bidders' belief on seller's honesty. Thus, it turns out that the signaling effect is stronger than the effect of manipulation. Unlike the discussion in related literature on the disadvantage of the application of the second-price auction in a setup with independently and identically distributed private values and a similar seller's manipulation (with different timing), our model favors the second-price auction

On the genericity of full surplus extraction in mechanism design

Heifetz and Neeman [A. Heifetz, Z. Neeman, On the generic (im)possibility of full surplus extraction, Econometrica 74 (2006) 213-233], using convex combinations of models, showed that full surplus extraction (FSE) in mechanism design is generically impossible, contrary to the seminal work of Cremer and McLean [J. Cremer, R. McLean, Full extraction of the surplus in Bayesian and dominant strategy auctions, Econometrica 53 (1988) 345-361]. Since Cremer and McLean did not allow convex combinations of models, the two results are not comparable. We show that FSE is generically impossible when convex combinations of models are not allowed, provided that we do not hold fixed the cardinality of models.Full surplus extraction Mechanism design Universal beliefs space Common knowledge

The genericity of beliefs-determine-preferences models revisited

Barelli [P. Barelli, On the genericity of full surplus extraction in mechanism design, J. Econ. Theory 144 (2009) 1320-1332] defines beliefs-determine-preferences (BDP) models and argues that BDP models are nongeneric in a topological sense. In this note, we point out some difficulties in Barelli[modifier letter apostrophe]s approach. Furthermore, we follow the idea of Heifetz and Neeman [A. Heifetz, Z. Neeman, On the generic (im)possibility of full surplus extraction, Econometrica 74 (2006) 213-233] to propose a more relevant notion of BDP* model. We prove that BDP* models are generic, which is opposite to Barelli[modifier letter apostrophe]s conclusion.Mechanism design Full surplus extraction Beliefs-determine-preferences priors Common priors