Revenue Maximising Agendas for Sequential English Auctions Shaheen Fatima and Michael Wooldridge Department of Computer Science University of Liverpool Liverpool L69 7ZF, U.K. shaheen,mjw @csc.liv.ac.uk Nicholas R. Jennings School of Electronics and Computer Science University of Southampton Southampton SO17 1BJ, U.K. nrj@ecs.soton.ac.uk Abstract This paper analyzes the way an auctioneer can arrange a series of sequential English auctions to maximize the to- tal revenue. We consider the case in which heterogeneous common value objects are sold in these auctions, and each bidder has a budget constraint that allows it to buy at most one object. In such a setting, bidders need to determine how much to bid in each auction. Furthermore, bidders’ strate- gies change if the agenda (i.e., the order in which the ob- jects are auctioned) changes. Consequently, the total rev- enue from the series of auctions can be changed by chang- ing the agenda. Given this, the auctioneer clearly wants to determine the optimal agenda (i.e., the one that maximizes the total revenue). To this end, we first determine the equi- librium bidding strategies for players that know their own budget constraint, but have incomplete information about the others’ constraints. On the basis of these strategies, we determine the total revenue for different agendas. 1. Introduction Multiple objects can be auctioned in two main ways: by us- ing combinatorial auctions or by auctioning each object in- dependently in a series of auctions (which can be arranged to occur simultaneously or which can be ordered sequen- tially). In this paper we focus on the latter approach and, in particular, on the sequential case. In this work we specifically consider the case where the bidders have budget constraints (i.e., a bidder’s budget does not allow it to buy more than one object). Thus, a bidder needs to determine how much to bid in each auction to max- imize its payoff across the complete set of auctions. For ex- ample, in sequential auctions for oil exploration rights, the price an oil company will pay for a particular area is affected not only by the area that is available in the current round, but also by the areas that will become available in subsequent rounds of leasing. Thus, it would be foolish for a bidder to spend all the money set aside for exploration on the first round of leasing, if potentially even more favourable sites are likely to be auctioned off subsequently. In other words, a bidder must determine bidding strategies that specify how much to bid in each individual auction. This decision mak- ing, and consequently the equilibrium outcome (i.e., the to- tal revenue from all the auctions), depends on the auction agenda (i.e., the order in which the auctions are conducted). Given this, the auctioneer wants to determine the optimal agenda (i.e., the one that maximizes total revenue) for con- ducting the auctions. In more detail, we consider the case where multiple het- erogeneous common value objects are sold in a series of En- glish auctions, one for each object. Furthermore, each bid- der can buy at most one object. We first determine the equi- librium strategies and then, on the basis of these strategies, we determine the revenue for each possible agenda. Most of the existing work in sequential auctions has fo- cused on multiple identical objects [5, 3]. The case of het- erogeneous private-value objects has been studied in [4], and common-value objects are considered in [1], but both make the complete information assumption. Elmaghraby [2] studies sequential auctions for heterogeneous private- value objects, using second-price sealed-bid rules, which complements our focus on common-value case. Against this background, this work makes three important contributions to the state of the art in multi-object auctions. First, it anal- yses the bidding behaviour of budget-constrained agents for sequential auctions for common-value objects in an incom- plete information setting. Second, it identifies the condi- tions under which it is possible for the auctioneer to deter- mine the optimal agenda on the basis of the available infor- mation. Finally, it determines the seller’s optimal agenda. 2. Sequential auctions There are two common-value objects and for sale and there are bidders ( ). All players are risk neu- tral expected profit maximizers. The seller and all bid- Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. AAMAS'04, July 19-23, 2004, New York, New York, USA. Copyright 2004 ACM 1-58113-864-4/04/0007...$5.00