No abstract

We study an aggregation problem in which a society has to determine its position on each of several issues, based on the positions of the members of the society on those issues. There is a prescribed set of feasible evaluations, i.e., permissible combinations of positions on the issues. The binary case of this problem, where only two positions are allowed on each issue, is by now quite well understood. We consider arbitrary sets of conceivable positions on each issue. This general framework admits the modeling of aggregation of various types of evaluations, including: assignments of candidates to jobs, choice functions from sets of alternatives, judgments in many-valued logic, probability estimates for events, etc. We require that the aggregation be performed issue-by-issue, and that the social position on each issue be supported by at least one member of the society. The set of feasible evaluations is called an impossibility domain if these requirements are satisfied for it only by dictatorial aggregation; that is to say, if it gives rise to an analogue of Arrow's impossibility theorem for preference aggregation. We obtain a two-part sufficient condition for an impossibility domain, and show that the major part is a necessary condition. For the ternary case, where three positions are allowed on each issue, we get a full characterization of impossibility domains.

We study strategyproof (SP) mechanisms for the location of a facility on a discrete graph. We give a full characterization of SP mechanisms on lines and on sufficiently large cycles. Interestingly, the characterization deviates from the one given by Schummer and Vohra [2004] for the continuous case. In particular, it is shown that an SP mechanism on a cycle is close to dictatorial, but all agents can affect the outcome, in contrast to the continuous case. Our characterization is also used to derive a lower bound on the approximation ratio with respect to the social cost that can be achieved by an SP mechanism on certain graphs. Finally, we show how the representation of such graphs as subsets of the binary cube reveals common properties of SP mechanisms and enables one to extend the lower bound to related domains.

No abstract

No abstract