JOURNAL OF ECONOMIC THEORY 34, 1-12 (1984) Davis-Hinich Conditions and Median Outcomes in Probabilistic Voting Models PETER J. COUGHLIN* Department of Economics, University of Maryland, College Park, Maryland 20742 Received July 8, 1982; revised April 12, 1984 Recent work has established that, in general, median voter results do not hold when there is probabilistic voting. This has, quite naturally, led to the following question: When, under such circumstances, will median outcomes occur? In this paper, it is shown that directly generalizing certain conditions that have previously been shown to be sufficient for multidimensional median voter results in deter- ministic voting models leads to conditions that are, themselves, sufftcient for median outcomes in probabilistic voting models. This generalization includes both discrete and continuous distributions of voters. In addition, it also applies to both unidimensional and multidimensional policy spaces. Journal of Economic Literature Classification Numbers: 025. 024, 022, 213, 320. ‘c 1984 Academic Press. inc. 1. INTR~OUCTI~N The first economic models of elections had candidates and/or policies identified with points in a undimensional set of alternatives (in particular, see Black [3,4] and Downs [15]). When these models were analyzed, they revealed the following basic fact: When their assumptions (including single- peaked preferences) are satisfied, any point which is a median for the distribution of voter ideal points cannot be defeated under majority rule (also see Arrow [ 1I). In the case of a two-candidate competition (where the premise of the previous sentenceis satisfied), this meansthat whenever both candidates have median positions they are in a pure strategy equilibrium. For obvious reasons, these results have come to be known over the years as “median voter results.” Through the work of a number of different researchers, it has also been learned that there are certain circumstances under which these unidimensional median voter results will have coun- terparts when candidates and/or policies are identified with points in multidimensional spaces.The pioneering work on this appeared in Davis and * I would like to acknowledge helpful comments and suggestions provided by Kenneth Arrow, Otto Davis, and Melvin Hinich. I would also like to acknowledge financial support provided by the Offtce of Naval Research (at Stanford University) and a Social Sciences and Public Policy Postdoctoral Fellowship (at Carnegie-Mellon University). 1 0022-053 l/84 $3.00 Copyright 0 1984 by Academic Press. Inc. All rights of reproduction in any form reserved.