A Naïve Bidder in a Common Value Auction

We study a common value auction in which two bidders compete for an item the value of which is a function of three independent characteristics. Each bidder observes one of these characteristics, but one of them is 'naive' in the sense that he does not realize the other bidder's signal contains useful information about the item's value. Therefore, this bidder bids as if this were an Independent Private Values auction. We show that the naive's bidder payoff exceeds that of his fully rational opponent for all symmetric unimodal signal distributions. We also show that naive bidding is persistent in the evolutionary sense.

The Greedy Heuristic Applied to a Class of Set Partitioning and Subset Selection Problems

The greedy heuristic may be used to obtain approximate solutions to integer programming problems. For some classes of problems, notably knapsack problems related to the coin changing problem, the greedy heuristic results in optimal solutions. However, the greedy heuristic does quite poorly at maximizing submodular set functions. This paper considers a class of set partitioning and subset selection problems. Results similar to those for maximizing submodular set functions are obtained for less restricted objective functions. The example used to show how poorly the heuristic does is motivated by a problem arising from an actual auction; the negative results are not mere mathematical pathologies but genuine shortcomings of the greedy heuristic. The greedy heuristic is quite successful at solving a class of knapsack problems related to the coin changing problem. Chang and Korsh [2], Hu and Lenard [5], Johnson and Kernighan [7], and Magazine, Nemhauser, and Trotter [8] show that the greedy heuristic results in optimal solutions for such problems. Problems of optimal subset selection have been studied by Boyce, Farhi, and Weischedel [1], indicating the need for a simply heuristic for obtaining approximate solutions. Fisher, Nemhauser, and Wolsey [4, 9, 10] have shown that the greedy heuristic may result in a solution for problems of maximizing submodular set functions with a value which is a relatively small fraction of the optimum. This paper derives similar results for a wider class of set partitioning and subset selection problems. The problem is formulated in the first section of the paper. Although the motivating problem results in a set partitioning problem, the results of the later sections apply as well to a wider class of subset selection problems. The more general problem statement is given as problem II; however, most of the discussion uses examples from the more restrictive problem I. The second section considers various possible restrictions to be placed on the objective function. The conditions may be stated in terms of either of the problem statements; the two forms of the conditions are shown to be essentially equivalent. Included among the possibilities are submodular set functions and several alternatives which are relaxations of submodularity. The relative generality of the various possibilities is illustrated by a couple of simple examples. The next two sections contain the main results of the paper. Objective functions which are “normal,” “monotonic,” and “discounted” are considered first. For such cases, the greedy heuristic solution is shown to have a value of at least 1/m of the optimal value, where m is the cardinality of the largest feasible subsets. The third section concludes by presenting a class of examples for which the greedy solution value is arbitrarily little more than the bound established above. Similar bounds may be obtained if the “discounted” condition is replaced by “variably discountedness,” although now the bounds must be functions of the variable discounting functions. Again, a lower bound is derived for the greedy solution value. The section concludes by presenting a class of examples for which the greedy solution value is arbitrarily little more than this bound. The last section is an attempt to reassure the reader that the above results are not simply pathological cases. An actual real estate auction [6] is briefly described. This real world problem is used to motivate bidding functions (of two hypothetical bidders) similar to those used to establish the tightness of the bound in sections three and four. This discussion suggests that the results are not mere mathematical pathologies and that, from many a practical viewpoint, the greed heuristic is not a satisfactory algorithm for obtaining optimal solutions to set partitioning and subset selection problems

Auctions and Bidding Models: A Survey

Auctions and bidding models are attracting an ever increasing amount of attention. The Stark and Rothkopf (1977) bibliography includes approximately 500 papers on the subject; additional work has been reported since the bibliography was compiled. This paper presents a general framework for classifying and describing various auctions and bidding models, and surveys the major results of the literature in terms of this framework

Sealed Bid Auctions with Non-Additive Bid Functions

A traditional sealed bid auction of a single item sells the item at the high bid price to a bidder with the highest bid. Such an auction may be used to auction several items; each bidder submits a bid on each item and each item is sold to a high bidder on that item. Implicit in this traditional scheme is the assumption that the bid for a set of items is the sum of the bids on the individual items: there are instances where this restriction appears unreasonable. This paper considers a more general sealed bid auction in which bids are submitted on all possible subsets of the items. The items are partitioned among the bidders to maximize revenue, where each bidder are partitioned among the bidders to maximize revenue, where each bidder pays what was bid on the set of items actually received. In general, the set partitioning problem is an extremely difficult integer programming problem, and there are two alternatives. The “greedy” and “sequential auction” heuristics are shown to result, at least for some examples, in very sub-optimal solutions. However, a class of slightly less general auction problems is presented for which optimal solutions may be calculated relatively easily; suggesting that some form of general sealed bid auctions may be appropriate in some situations

Auctions and Bidding Models: A Survey

Auctions and bidding models are attracting an ever increasing amount of attention. The Stark and Rothkopf (1977) bibliography includes approximately 500 papers on the subject; additional work has been reported since the bibliography was compiled. This paper presents a general framework for classifying and describing various auctions and bidding models, and surveys the major results of the literature in terms of this framework

A Model for the Distribution of the Number of Bidders in an Auction

The distribution of the number of bidders in auctions with uncertain numbers is usually assumed to be Poisson. The observed distribution, for example in OCS Federal Offshore Oil Lease Sales, is apparently not Poisson. A simple model is presented showing that if the objects have different values and individuals tend to only bid on objects with high value, then the resulting distribution of the number of bidders will not be Poisson. The results of the model correspond closely to data observed in Federal Offshore Oil lease auctions and the model is simple enough so that it may be of practical use to an individual participating in such an auction

Multiplicative Bidding and Convergence to Equilibrium

General equilibrium strategies may be relatively difficult to determine. While multiplicative strategies may be much simpler to calculate, they are not in general in equilibrium. An example shows that such strategies may indeed be quite far from equilibrium. However, if the example is iterated using Bayesian decision analysis, the strategies quickly converge to being very nearly in equilibrium

An Example of a Multi-Object Auction Game

Multi-object auctions are traditionally analyzed as if they were a number of simultaneous independent single object auctions. Such an approximation may be very crude if bidders have budget restrictions, capacity constraints, or, in general, have non-linear utility functions. This paper presents a very simple multi-object auction for which explicit equilibrium strategies can be calculated; these equilibrium strategies have several qualitative characteristics arising from the multi-object nature of the example and therefore not present in typical single object auctions

Bidding in Auctions with Multiplicative Lognormal Errors: An Example

Auction models with lognormally-distributed multiplicative errors are used extensively in models of mineral lease sales. Equilibrium strategies are typically difficult to calculate; multiplicative strategies are often used as approximations. An example based on a federal offshore oil lease sale shows that multiplicative strategies may be quite far from being in equilibrium. However, under a special form of repetition, such strategies converge very rapidly to an equilibrium. The effects of any fixed costs, the reservation price, the number of bidders and the variance of the error are examined briefly for this example