A Note on Revenue Effects of Asymmetry in Private-Value Auctions

We formulate a way to study whether the asymmetry of buyers (in the sense of having different prior probability distributions of valuations) is helpful to the seller in private-value auctions (asked first by Cantillon [2001]). In our proposed formulation, this question corresponds to two important questions previously asked: Does a first-price auction have higher revenue than a second-price auction when buyers have asymmetric distributions (asked by Maskin and Riley[2000])? And does a seller enhance revenue by releasing information (asked by Milgrom and Weber[1982])? This is shown by constructing two Harsanyi games of incomplete information each having the same ex-ante distribution of valuations but in one beliefs are symmetric while in the other beliefs are sometimes asymmetric. Our main result is that answers to all three questions coincide when values are independent and are related when values are affiliated.Asymmetric Auctions; Asymmetric Beliefs; Affiliation; Linkage Principle

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

A Sealed-Bid Auction that Matches the English Auction

This paper analyzes a two-stage sealed-bid auction that is frequently employed in privatization, takeover, and merger and acquisition contests. This auction format yields the same expected revenue as the open ascending (English) auction, yet is less susceptible to preemptive bidding and collusion.Auctions

Positive value of information in games.

We exhibit a general class of interactive decision situations in which all the agents benefit from more information. This class includes as a special case the classical comparison of statistical experiments a la Blackwell.Information structures, value of information, Pareto optima.

Asymmetric first-price auctions with uniform distributions: Analytic solutions to the general case

Draft version dated December 25, 2007 originally deposited in SSRN working paper series. Final version published by Springer; available online at http://www.springer.com/While auction research, including asymmetric auctions, has grown significantly in recent years, there is still little analytical solutions of first-price auctions outside the symmetric case. Even in the uniform case, Griesmer et al. (1967) and Plum (1992) find solutions only to the case where the lower bounds of the two distributions are the same. We present the general analytical solutions to asymmetric auctions in the uniform case for two bidders, both with and without a minimum bid. We show that our solution is consistent with the previously known solutions of auctions with uniform distributions. Several interesting examples are presented including a class where the two bid functions are linear. We hope this result improves our understanding of auctions and provides a useful tool for future research in auctions

Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle

The fibers in a skeletal muscle are divided into groups called “muscle units” whereby each muscle unit is innervated by a single neuron. It was found that neurons with low activation thresholds have smaller muscle units than neurons with higher activation thresholds. This results in a fixed recruitment order of muscle units, from smallest to largest, called the “size principle.” It is thought that the size principle results from a competitive process—taking place after birth—between the neurons innervating the muscle. The underlying mechanism of the competition was not understood. Moreover, the results in the majority of experiments that manipulated the activity during the competition period seemed to contradict the size principle. Experiments at the isolated muscle fibers showed that the competition is governed by a Hebbian-like rule, whereby neurons with low activation thresholds have a competitive advantage at any single muscle fiber. Thus neurons with low activation thresholds are expected to have larger muscle units in contradiction to what is seen empirically. This state of affairs was termed “paradoxical.” In the present study we developed a new game theoretic framework to analyze such competitive biological processes. In this game, neurons are the players competing to innervate a maximal number of muscle fibers. We showed that in order to innervate more muscle fibers, it is advantageous to win (as the neurons with higher activation thresholds do) later competitions. This both explains the size principle and resolves the seemingly paradoxical experimental data. Our model establishes that the competition at each muscle fiber may indeed be Hebbian and that the size principle still emerges from these competitions as an overall property of the system. Thus, the less active neurons “lose the battle but win the war.” Our model provides experimentally testable predictions. The new game-theoretic approach may be applied to competitions in other biological systems