How many candidates are needed to make elections hard to manipulate?

In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. The complexity of manipulating realistic elections where the number of candidates is a small constant was recently studied (Conitzer 2002), but the emphasis was on the question of whether or not a protocol becomes hard to manipulate for some constant number of candidates. That work, in many cases, left open the question: How many candidates are needed to make elections hard to manipulate? This is a crucial question when comparing the relative manipulability of different voting protocols. In this paper we answer that question for the voting protocols of the earlier study: plurality, Borda, STV, Copeland, maximin, regular cup, and randomized cup. We also answer that question for two voting protocols for which no results on the complexity of manipulation have been derived before: veto and plurality with runoff. It turns out that the voting protocols under study become hard to manipulate at 3 candidates, 4 candidates, 7 candidates, or never

Vertex-reinforced random walk on Z eventually gets stuck on five points

Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the one-dimensional integer lattice Z. They proved that the range is almost surely finite and that with positive probability the range contains exactly five points. They conjectured that this second event holds with probability 1. The proof of this conjecture is the main purpose of this paper.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790700000069

Operator-valued spectral measures and large deviations

International audienceWe study large deviation properties of random matricial spectral measures

Expressive Power of Weighted Propositional Formulas for Cardinal Preference Modelling

As proposed in various places, a set of propositional formulas, each associated with a numerical weight, can be used to model the preferences of an agent in combinatorial domains. If the range of possible choices can be represented by the set of possible assignments of propositional symbols to truth values, then the utility of an assignment is given by the sum of the weights of the formulas it satisfies. Our aim in this paper is twofold: (1) to establish correspondences between certain types of weighted formulas and well-known classes of utility functions (such as monotonic, concave or k-additive functions); and (2) to obtain results on the comparative succinctness of different types of weighted formulas for representing the same class of utility functions