Optimal Design Of English Auctions With Discrete Bid Levels

This paper considers a form of ascending price English auction widely used in both live and online auctions. This discrete bid auction requires that the bidders submit bids at predetermined discrete bid levels, and thus, there exists a minimal increment by which the bid price may be raised. In contrast, the academic literature of optimal auction design deals almost solely with continuous bid auctions. As a result, there is little practical guidance as to how an auctioneer, seeking to maximize its revenue, should determine the number and value of these discrete bid levels, and it is this omission that is addressed here. To this end, a model of a discrete bid auction from the literature is considered, and an expression for the expected revenue of this auction is derived. This expression is used to determine both numerical and analytical solutions for the optimal bid levels, and uniform and exponential bidder’s valuation distributions are compared. Finally, the limiting case where the number of discrete bid levels is large is considered. An analytical expression for the distribution of the optimal discrete bid levels is derived, and an intuitive understanding of how this distribution maximizes the revenue of the auction is developed

Optimal design of Dutch auctions with discrete bid levels.

The theory of auction has become an active research area spanning multiple disciplines such as economics, finance, marketing and management science. But a close examination of it reveals that most of the existing studies deal with ascending (i.e., English) auctions in which it is assumed that the bid increments are continuous. There is a clear lack of research on optimal descending (i.e., Dutch) auction design with discrete bid levels. This dissertation aims to fill this void by considering single-unit, open-bid, first price Dutch auctions in which the bid levels are restricted to a finite set of values, the number of bidders may be certain or uncertain, and a secret reserve price may be present or absent. These types of auctions are most attractive for selling products that are perishable (e.g., flowers) or whose value decreases with time (e.g., air flight seats and concert tickets) (Carare and Rothkopf, 2005). I began by conducting a comprehensive survey of the current literature to identify the key dimensions of an auction model. I then zeroed in on the particular combination of parameters that characterize the Dutch auctions of interest. As a significant departure from the traditional methods employed by applied economists and game theorists, a novel approach is taken by formulating the auctioning problem as a constrained mathematical program and applying standard nonlinear optimization techniques to solve it. In each of the basic Dutch auction model and its two extensions, interesting properties possessed by the optimal bid levels and the auctioneer's maximum expected revenue are uncovered. Numerical examples are provided to illustrate the major propositions where appropriate. The superiority of the optimal strategy recommended in this study over two commonly-used heuristic procedures for setting bid levels is also demonstrated both theoretically and empirically. Finally, economic as well as managerial implications of the findings reported in this dissertation research are discussed

Sealed bid second price auctions with discrete bids

A single item is sold to two bidders by way of a sealed bid second price auction in which bids are restricted to a set of discrete values. Restricting attention to symmetric pure strategy behavior on the part of bidders, a unique equilibrium exists. When following these equilibrium strategies bidders may bid strictly above or below their valuation, implying that the item may be awarded to a bidder other than the high valuation bidder. In an auction with two acceptable bids, the expected revenue of the seller may be maximized by a high bid level not equal to the highest possible bidder valuation and may exceed the expected revenue from an analogous second price auction with continuous bidding (and no reserve price). With three acceptable bids, a revenue maximizing seller may choose unevenly spaced bids. With an arbitrary number of evenly spaced bids, as the number of acceptable bids is increased, the expected revenue of the seller and the probability of ex post inefficiency both may either increase or decrease

A communication equilibrium in English auctions with discrete bidding

This paper analyses a model of a common value English auction with discrete bidding. In this model, we show that there exists a communication equilibrium in which the high signal bidder strategically chooses his first bid so as to maximise his expected utility. Straightforward bidding, or increasing the bid by the minimum amount possible, is the equilibrium strategy for both bidders in all other auction rounds. We relate this result to recent research on English auctions with discrete bidding and auctions where bidders may have noisy information about their opponent's signals.English Auctions, discrete bidding, communication equilibrium

Discrete Clock Auctions: An Experimental Study

We analyze the implications of different pricing rules in discrete clock auctions. The two most common pricing rules are highest-rejected bid (HRB) and lowest-accepted bid (LAB). Under HRB, the winners pay the lowest price that clears the market; under LAB, the winners pay the highest price that clears the market. Both the HRB and LAB auctions maximize revenues and are fully efficient in our setting. Our experimental results indicate that the LAB auction achieves higher revenues. This also is the case in a version of the clock auction with provisional winners. This revenue result may explain the frequent use of LAB pricing. On the other hand, HRB is successful in eliciting true values of the bidders both theoretically and experimentally.Auctions, clock auctions, spectrum auctions, experimental economics, behavioral economics, market design

Auctions with Severely Bounded Communication

We study auctions with severe bounds on the communication allowed: each bidder may only transmit t bits of information to the auctioneer. We consider both welfare- and profit-maximizing auctions under this communication restriction. For both measures, we determine the optimal auction and show that the loss incurred relative to unconstrained auctions is mild. We prove non-surprising properties of these kinds of auctions, e.g., that in optimal mechanisms bidders simply report the interval in which their valuation lies in, as well as some surprising properties, e.g., that asymmetric auctions are better than symmetric ones and that multi-round auctions reduce the communication complexity only by a linear factor

Fraction auctions: the tradeoff between effciency and running time

This paper studies the sales of a single indivisible object where bidders have continuous valuations. In Grigorieva et al. [13] it was shown that, in this setting, query auctions necessarily allocate inefficiently in equilibrium. In this paper we propose a new sequential auction, called the c-fraction auction. We show c-fraction auctions guarantee approximate efficiency at any desired level of accuracy, independent of the number of bidders. We discuss the running time and the efficiency in the ex-post equilibrium of the auction. We show that by changing the parameter c of the auction we can trade off efficiency against running time.operations research and management science;

Experimental Evidence on English Auctions: Oral Outcry vs. Clock

This paper tests experimentally, in a common value setting, the equivalence between the Japanese English auction (or clock auction) and an open outcry auction, where bidders are allowed to call their own bids. We find that (i) bidding behaviour is different in each type of auction, but also that (ii) this difference in bidding behaviour does not affect significantly the auction prices. This lends some support to the equivalence between these two types of auction. The winner's curse is present: overbidding led to higher than expected prices (under Nash bidding strategies) in both types of auction.English auctions, discrete bidding, winner's curse

An Investigation Report on Auction Mechanism Design

Auctions are markets with strict regulations governing the information available to traders in the market and the possible actions they can take. Since well designed auctions achieve desirable economic outcomes, they have been widely used in solving real-world optimization problems, and in structuring stock or futures exchanges. Auctions also provide a very valuable testing-ground for economic theory, and they play an important role in computer-based control systems. Auction mechanism design aims to manipulate the rules of an auction in order to achieve specific goals. Economists traditionally use mathematical methods, mainly game theory, to analyze auctions and design new auction forms. However, due to the high complexity of auctions, the mathematical models are typically simplified to obtain results, and this makes it difficult to apply results derived from such models to market environments in the real world. As a result, researchers are turning to empirical approaches. This report aims to survey the theoretical and empirical approaches to designing auction mechanisms and trading strategies with more weights on empirical ones, and build the foundation for further research in the field

Learn While You Earn: Two Approaches to Learning Auction Parameters in Take-it-or-leave-it Auctions

Much of the research in auction theory assumes that the auctioneer knows the distribution of participants ’ valuations with complete certainty. However, this is unrealistic. Thus, we analyse cases in which the auctioneer is uncertain about the valuation distributions; specifically, we consider a repeated auction setting in which the auctioneer can learn these distributions. Using take-it-or-leave-it auctions (Sandholm and Gilpin, 2006) as an exemplar auction format, we consider two auction design criteria. Firstly, an auctioneer could maximise expected revenue each time the auction is held. Secondly, an auctioneer could maximise the information gained in earlier auctions (as measured by the Kullback-Liebler divergence between its posterior and prior) to develop good estimates of the unknowns, which are later exploited to improve the revenue earned in the long-run. Simulation results comparing the two criteria indicate that setting offers to maximise revenue does not significantly detract from learning performance, but optimising offers for information gain substantially reduces expected revenue while not producing significantly better parameter estimates