T here is interest in designing simultaneous auctions for situations such as the recent FCC radio spectrum auctions, in which the value of assets to a bidder depends on which other assets he or she wins. In such auctions, bidders may wish to submit bids for combinations of assets. When this is allowed, the problem of determining the revenue maximizing set of nonconflicting bids can be difficult. We analyze this problem, identifying several different structures of permitted combinational bids for which computational tractability is constructively demonstrated and some structures for which computational tractability cannot be guaranteed.

Combinatorial auctions have two features that greatly affect their design: computational complexity of winner determination and opportunities for cooperation among competitors. Dealing with these forces trade-offs between desirable auction properties such as allocative efficiency, revenue maximization, low transaction costs, fairness, failure freeness, and scalability. Computational complexity can be dealt with algorithmically by relegating the computational burden to bidders, by maintaining fairness in the face of computational limitations, by limiting biddable combinations, and by limiting the use of combinatorial bids. Combinatorial auction designs include single-round, first-price sealed bidding, Vickrey-Clarke-Groves (VCG) mechanisms, uniform and market-clearing price auctions, and iterative combinatorial auctions. Combinatorial auction designs must deal with exposure problems, threshold problems, ways to keep the bidding moving at a reasonable pace, avoiding and resolving ties, and controlling complexity.Auction Design, Combinatorial Bidding, Bidding with Synergies

Approval voting, probabilistic choice models, probabilistic ranking models, subset choice,

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This paper investigates algebraic and combinatorial properties of the set of linear orders on the algebra of subsets of a finite set that are representable by positive measures. It is motivated by topics in decision theory and the theory of measurement, where an understanding of such properties can facilitate the design of strategies to elicit comparisons between subsets that, for example, determine an individual's preference order over subsets of objects or an individual's qualitative probability order over subsets of states of the world. We introduce a notion of critical pairs of binary comparisons for such orders and prove that (i) each order is uniquely characterized by its set of critical pairs and (ii) the smallest set of binary comparisons that determines an order is a subset of its set of critical pairs. The paper then focuses on the minimum number of on-line binary-comparison queries between subsets that suffice to determine any representable order for a set of given cardinality n. It is observed that, for small n, the minimum is attained by first determining the ordering of singleton subsets. We also consider query procedures with fixed numbers of stages, in each stage of which a number of queries for the next stage are formulated.1. Introduction. This paper investigates algebraic and combinatorial properties of additive linear orders on the Boolean algebra n of subsets of an n-item set n = 1 2 n . We refer to a binary relation ≺ on n as an additive linear order if there exists a measure on n such that i = i > 0 for all i ∈ n , and with A = i∈A i for all A ∈ n ,

This essay assesses the state of auction theory in a particular dimension: its relevance to practice. Most auction models are more abstract than necessary. They depend on assumptions that are highly unlikely to occur in practical situations, which are often less formal and rigid. Nonetheless, we discuss several significant steps toward offering more practical advice.

We consider a 'Maker-Breaker' version of the Ramsey Graph Game, RG(n), and present a winning strategy for Maker requiring at most (« -3)2"~' + n + 1 moves. This is the fastest winning strategy known so far. We also demonstrate how the ideas presented can be used to develop winning strategies for some related combinatorial games.

consider the effect of changes of scale of measurement on the conclusion that a particular solution to a scheduling problem is optimal. The analysis in this paper was motivated by the problem of finding the optimal transportation schedule when there are penalties for both late and early arrivals, and when different items that need to be transported receive different priorities. We note that in this problem, if attention is paid to how certain parameters are measured, then a change of scale of measurement might lead to the anomalous situation where a schedule is optimal if the parameter is measured in one way, but not if the parameter is measured in a different way that seems equally acceptable. This conclusion about the sensitivity of the conclusion that a given solution to a combinatorial optimisation problem is optimal is different from the usual type of conclusion in sensitivity analysis, since it holds even though there is no change in the objective function, the constraints, or other input parameters, but only in scales of measurement. We emphasize the need to consider such changes of scale in analysis of scheduling and other combinatorial optimization problems. We also discuss the mathematical problems that arise in two special csses, where all desired arrival times are the same and the simplest case where they are not, namely the case where there are two distinct arrival times but one of them occurs exactly once. While specialized, these two examples illustrate the types of mathematical problems that arise from considerations of the interplay between scaletypes and optimisation.