Optimal Bidding Algorithms Against Cheating in Multiple-Object Auctions

This paper studies some basic problems in a multiple-object auction model using methodologies from theoretical computer science. We are especially concerned with situations where an adversary bidder knows the bidding algorithms of all the other bidders. In the two-bidder case, we derive an optimal randomized bidding algorithm, by which the disadvantaged bidder can procure at least half of the auction objects despite the adversary's a priori knowledge of his algorithm. In the general k-bidder case, if the number of objects is a multiple of k, an optimal randomized bidding algorithm is found. If the k − 1 disadvantaged bidders employ that same algorithm, each of them can obtain at least 1/k of the objects regardless of the bidding algorithm the adversary uses. These two algorithms are based on closed-form solutions to certain multivariate probability distributions. In situations where a closed-form solution cannot be obtained, we study a restricted class of bidding algorithms as an approximation to desired optimal algorithms. 1. Introduction. This paper investigates some basic problems in auction theory. Broadly speaking, an auction is a market mechanism with explicit or implicit rules for allocating resources and determining prices on the basis of bids from market participants [4, 10, 12, 18]. Auctions are frequently used to price various types of assets. For instance, the U.S. Treasury raises funds by auctioning T-bonds and T-notes, while the Department of the Interior sells mineral rights on federally owned properties via auction. Economists are interested in auctions as an efficient way to price and allocate goods which have no standard market value. Auctions are believed to be the simplest and most familiar means of price determination for multilateral trading without intermediary market makers [10, 12, 18]. In typical auctions, there are one seller and a group of competing buyers who bid to possess the auction objects. Procurements describe situations in which a single buyer wishes to purchase objects from a set of potential suppliers. There are four basic forms of auctions in use [10, 12, 14]. In an English auction or ascending bid auction, the price of an object is successively raised until only one bidder remains and wins the object. In a Dutch auction, which is the converse of an English auction, an initial high price is subsequently lowered until a bidder accepts the current price. In a first-price sealed-bid auction, potential buyers submit sealed bids for an object. The highest bidder is …