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

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;

A note on the wallet game with discrete bid levels

It is well-known that in the wallet game with two bidders, bidding twice the (individual) signal is an equilibrium. We prove that this strategy is never an equilibrium in a Japanese-English auction once discrete bid levels are introduced; we also discuss the implications of this result

Fraction auctions : the tradeoff between efficiency and running time

This paper studies the sales of a single indivisible object where bidders have continuous valuations. In grigorieva et al. [14] 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 the existence of an ex-post equilibrium, called bluff equilibrium, in which bidders behave truthfully except for particular constellations of observed bids at which it is optimal to pretend a slightly higher valuation. We show c-fraction auctions guarantee approximate efficiency at any desired level of accuracy, independent of the number of bidders, when bidders choose to play the bluff equilibrium. We discuss the running time and the efficiency in the bluff equilibrium. We show that by changing the parameter c of the auction we can trade off efficiency against running time

Inefficiency of equilibria in digital mechanisms with continuous valuations.

A digital mechanism is defined as an iterative procedure in which bidders select an action, from a finite set, in each iteration. When bidders have continuous valuations and make strategic reports, we show that any ex post implementation of the vickrey choice rule via such a mechanism needs infinitely many iterations for almost all realizations of the bidders’ valuations. Thus, when valuations are drawn from a continuous probability distribution, the vickrey choice rule can only be used at the expense of a running time that is infinite with probability one. This infeasibility result even holds in the case of two bidders and the vickrey choice rule only being required to be established with probability one. Establishing the efficient allocation when the nn bidders’ report truthfully contrasts starkly to the previous setting: a bisection procedure has a finite running time almost always, and an expected number of reports are equal to 2n2n. Using a groves payment scheme rather than vickrey’s second price payment scheme somewhat mitigates the problem. We provide an example mechanism with a groves payment scheme, in which the running time of the mechanism in equilibrium is finite with probability 1212

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

On The Fastest Vickrey Algorithm

We investigate the algorithmic performance of Vickrey-Clarke-Groves mechanisms in the single item case. We provide a formal definition of a Vickrey algorithm for this framework, and give a number of examples of Vickrey algorithms. We consider three performance criteria, one corresponding to a Pareto criterion, one corresponding to worst case analysis, and a third criterion related to first-order stochastic dominance. We show that Pareto optimal Vickrey algorithms do not exist and that worst case analysis is of no use in discriminating between Vickrey algorithms. For the case of two bidders, we show the bisection auction to be optimal according to the third criterion. The bisection auction istherefore optimal in a very strong sense.operations research and management science;