Richard Stong

Collective choice under dichotomous preferences

Agents partition deterministic outcomes into good or bad. A direct revelation mechanism selects a lottery over outcomes - also interpreted as time-shares. Under such dichotomous preferences, the probability that the lottery outcome be a good one is a canonical utility representation. The utilitarian mechanism averages over all deterministic outcomes approved by the largest number of agents. It is efficient, strategy-proof and treats equally agents and outcomes. We reach the impossibility frontier if we also place the lower bound 1/n on each agents utility, where n is the number of agents; or if this lower bound is the fraction of good outcomes to feasible outcomes. We conjecture that no ex-ante efficient and strategy-proof mechanism guarantees a strictly positive utility to all agents at all profiles, and prove a weaker version of this conjecture.

Fair Queuing and Other Probabilistic Allocation Methods

A server processes one job per unit of time and randomly schedules the jobs requested by a given set of users; each user may request a different number of jobs.Fair queuing (Shenker 1989) schedules jobs in successive round-robin fashion, where each agent receives one unit in each round until his demand is met and the ordering is random in each round. Fair queuing *, the reverse scheduling of fair queuing, serves first (with uniform probability) one of the users with the largest remaining demand.We characterize fair queuing * by the combination of lower composition--LC--(the scheduling sequence is history independent), demand monotonicity--DM--(increasing my demand cannot result in increased delay) and two equity axioms, equal treatment ex ante--ETEA (two identical demands give the same probability distribution of service) and equal treatment ex post--ETEP (two identical demands must be served in alternating fashion). The set of dual axioms (in which ETEA and ETEP are unchanged) characterizes fair queuing.We also characterize the rich family of methods satisfying LC, DM, and the familiar consistency--CSY--axiom. They work by fixing a standard of comparison (preordering) between a demand ofx iunits by agenti and one ofx junits by agentj. The first job scheduled is drawn from the agents whose demand has the highest standard.

The average order of a matrix

Abstract Let GL ( n , q ) denote the group of invertible n × n matrices over the finite field with q elements and let v n denote the average order of an element in this group. We show that for n large log v n = n log q − log n + o (log n ).

Filling a Multicolor Urn: An Axiomatic Analysis

We study the probabilistic distribution of identical successive units. We represent the allocation process as the filling of an urn with balls of different colors (one color per agent). Applications include the scheduling of homogeneous tasks among workers and allocating new workers between divisions. The fixed chances methods allocate each unit independently of the current distribution of shares. The Polya-Eggenberger methods place in an urn a fixed number of balls and draw from the urn with replacement of two balls of the color drawn. These two families of urn-filling methods emerge uniquely from our axiomatic discussion involving: a version of the familiar Consistency property; Share Monotonicity (my probability of receiving the next ball is non-decreasing in my current share); Independence of Transfers (transferring balls across agents is not profitable), and Order Independence (a sequence of successive allocations is as likely as any permuted sequence). We also explore the impact of Share Monotinicity (my probability of receiving the next ball is non-increasing in my current share), leading to an equalization of individual shares along a fixed standard of comparison.

Orthogonal matchings

If a graph G is decomposed into m 2-factors, then an orthogonal matching is a matching M in G which contains exactly one edge from each 2-factor. It has been conjectured that any 2-factorization of any graph has an orthogonal matching. We prove this conjecture under the additional assumption that G has at least 3m - 2 vertices.

Self-homeomorphisms of 4-manifolds with fundamental group Z

Abstract In this paper we study the classification of self-homeomorphisms of closed, connected, oriented 4-manifolds with infinite cyclic fundamental group up to pseudoisotopy, or equivalently up to homotopy. We find that for manifolds with even intersection form homeomorphisms are classified up to pseudoisotopy by their action on π1 , π2 and the set of spin structures on the manifold. For manifolds with odd intersection form they are classified by the action on π1 and π2 and an additional Z /2 Z . As a consequence we complete the classification program for closed, connected, oriented 4-manifolds with infinite cyclic fundamental group, begun by Freedman, Quinn and Wang.

Uniqueness of π1-negligible embeddings in 4-manifolds: A correction to theorem 10.5 of Freedman and Quinn

The purpose of this note is to provide a correction to the existence part of Theorems 10.3 and 10.5 of Topology of 4-manifolds, Princeton Univ. Press, Princeton, NJ, 1990, which analyze when one can find a connected sum decomposition of a 4-manifold or a 71-negligible embedding in a 4-manifold respectively. In particular this gives a correction to the definition of the 4-dimensional Kervaire-Milnor invariant. We also define this invariant in a slightly more general context.

Buchsbaum and Eulerian complexes

Let Δ be a finite simplicial complex, and K [Δ] its Stanley-Reisner ring. We show that if Δ is Eulerian, then every depth of K[Δ] is possible, and we find all possible depths of K [Δ] for Buchsbaum Eulerian Δ, dealing with the orientable and nonorientable cases separately. In addition, we find all possible betti sequences of Buchsbaum-Eulerian, Eulerian, and semi-Eulerian complexes.

Uniqueness of connected sum decompositions in dimension 4

Abstract Suppose W is a 4-manifold with good fundamental group and M is a closed simply-connected 4-manifold. Suppose we are given two decompositions h 1 : W ⋍ M # W 1 and h 2 : W ⋍ M # W 2 inducing the same decomposition of π 2 W . In this paper we study when we can conclude that W 1 and W 2 are homeomorphic. As a consequence we conclude that the ∗ operation for changing the Kirby-Siebenmann invariant of a 4-manifold is well defined. We will also use this discussion to relate the ambient approach to classification to the surgery approach.

Measured lamination spaces on surfaces and geometric intersection numbers

Abstract In this paper, we produce an elementary approach to Thurstons theory of measured laminations on compact surfaces with non-empty boundary. We show that the theory can be derived from a simple inequality for geometric intersection numbers between arcs inside an octagon.