Combinatorial Auctions with Decreasing Marginal Utilities

In most of microeconomic theory, consumers are assumed to exhibit decreasing marginal utilities. This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valuations is NP-hard, we present an efficient greedy 2-approximation algorithm for this case and generalize it to the case of limited complementarities. No such approximation algorithm exists in a setting allowing for arbitrary complementarities. Some results about strategic aspects of combinatorial auctions among players with decreasing marginal utilities are also presented.Comment: To appear in GEB. Preliminary version appeared in EC'0

A presentation of Quantum Logic based on an "and then" connective

When a physicist performs a quantic measurement, new information about the system at hand is gathered. This paper studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the "and then" operation that combines old and new information. The "and then" connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the "and then" connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch (1969) satisfies all the properties studied in this paper. All properties satisfied by Finch's operation in modular lattices are valid in Hilbert Space Quantum Logic. It is not known whether all properties of Hilbert Space Quantum Logic are satisfied by Finch's operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.Comment: 28 pages. Submitte

An Apparent Order of Sensory Ability Changes in Human Beings \ud

Examination of records and personal documents of 1309 human beings isolated an apparent order of sensory ability changes, for 154 members of the sample both records and personal documents were available. The five changes in the apparent order are (a) an improvement in auditory perception that always includes an increase in complexity of speech, (b) an improvement in ability to taste or to smell or both and comparative myopia in which the developing human being becomes more near-sighted than he or she has been, (c) an increase in ability to discern and separate aural or visual or aural and visual stimuli simultaneously received, (d) comparative hyperopia, and (e) a marked increase in ability in one or more up to all five of the sensory abilities considered in the research: audition, vision, gustation, olfaction, and touch. Truncation of movement through the order is a salient feature, and the fourth change, comparative hyperopia, may be skipped. If the apparent order is significant and supportable, it may lead to new, productive research in many affected disciplines

Iitaka dimension for cycles

We define the Iitaka dimension of a numerical cycle class and develop its theory. We conjecture that the Iitaka dimension is integer-valued, and give some evidence in this direction. We focus on two cases of geometric interest: Schubert cycles on Grassmannians and cycles contracted by morphisms.Comment: 24 page

Algebraic bounds on analytic multiplier ideals

Given a pseudo-effective divisor L we construct the diminished ideal of L, a "continuous" extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier ideal of the metric of minimal singularities of L is contained in the diminished ideal. We also characterize abundant divisors using the diminished ideal, indicating that in this case the geometric and analytic information should coincide.Comment: 27 pages; v3: corrected several mistake