Abstract
Niemi (1969), in an important but neglected paper, found that when orderings were drawn from a simulation based on the impartial culture, the greater the proportion of voter orderings that were single-peaked (a condition he called partial single-peakedness), the more likely was there to be a transitive group ordering. Niemi also found that the likelihood of transitivity increased with n, group size — approaching one as n grew large. Niemi's simulation was restricted to the case of three alternatives. Also, he provided no theoretical explanation for the results of his simulation. Here we provide a theoretical explanation for Niemi's results in terms of a model based on the idea of net preferences, and we extend his results for the general case of any finite number of alternatives, m, for electorates that are large relative to the number of alternatives being considered. In addition to providing a rationale for Niemi's (1969) simulation results, the ideas of net preferences and opposite preference we make use of have a wide range of potential applications.
This is a preview of subscription content, log in via an institution to check access.
References
Arrow, K.J. (1963). Social choice and individual values, 2nd Edition. New York: Wiley.
Bell, C.E. (1978). What happens when majority rule breaks down: Some probability calculations. Public Choice 33(2): 121–126.
Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press.
Condorcet, Marquis de (1985). Essai sur l'application de l'analyse à la probabilité des decisions rendues à la pluralité des voix. Paris.
Gaertner, W., and Heinecke, A. (1978). Cyclically mixed preferences: A necessary and sufficient condition for transitivity of the social preference relation. In H.W. Gottinger and W. Leinfellner, Decision theory and social ethnics: Issues in social choice, 169–186. Dordrecht: Reidel.
Gehrlein, W.V., and Fishburn, P.C. (1976). The probability of the paradox of voting: A computable solution. Journal of Economic Theory 13(1) (August): 14–25.
Kramer, G.H. (1973). On a class of equilibrium conditions for majority rule. Econometrica 41: 285–297.
Kuga, K., and Nagatani, H. (1974). Voter antagonism and the paradox of voting. Econometrica 42(6) (November): 1045–1067.
McKelvey, R.D. (1976). Intransitivities in multidimensional voting models and some implications for agenda control. Journal of Economic Theory 12: 472–482.
McKelvey, R., and Wendell, R. (1976). Voting equilibria in multidimensional choice spaces. Mathematics of Operations Research 1: 144–158.
Miller, N. (1970). A necessary and sufficient condition for transitive majority orderings. Unpublished manuscript.
Nicholson, M.B. (1965). Conditions for the ‘voting paradox’ for committee decisions. Metroeconomica 7: 29–44.
Niemi, R.G. (1969). Majority decision-making with partial unidimensionality. American Political Science Review 63: 488–497.
Niemi, R.G. (1970). The occurrence of the paradox of voting in university elections. Public Choice 8: 91–100.
Niemi, R.G. (1983). Why so much stability?! Another opinion. Public Choice 41: 261–270.
Niemi, R.G., and Weisberg, H.F. (1968). A mathematical solution for the probability of the paradox of voting. Behavioral Science 13: 317–323.
Plott, C. (1967). A notion of equilibrium and its possibility under major rule. American Economic Review 57: 787–806.
Riker, W.H. (1961). Voting and the summation of preferences: An interpretive bibliographic essay of selected developments during the last decade. American Political Science Review 55: 900–911.
Riker, W. (1980). Implications from the disequilibrium of majority rule for the study of institutions. American Political Science Review 74(1): 432–446, and Reply, 456–458.
Riker, W. (1982). Liberalism against populism: A confrontation between the theory of democracy and the theory of social choice. San Francisco: Freeman.
Schofield, N. (1978). Instability of simple dynamic games. Review of Economic Studies 45(3): 575–594.
Sen, A.K. (1966). A Possibility theorem on majority decisions. Econometrica 34: 491–499.
Sen, A.K. Social choice theory. In K.J. Arrow and M. Intriligator (Eds.), Handbook of mathematical economics, forthcoming.
Author information
Authors and Affiliations
Additional information
We are indebted to Jonathan Riley for helpful bibliographic assistance, and to Sue Pursche and the staff of the Word Processing Center, School of Social Sciences, U.C.I. for typing several earlier drafts of this manuscript.
Rights and permissions
About this article
Cite this article
Feld, S.L., Grofman, B. Research note Partial single-peakedness: An extension and clarification. Public Choice 51, 71–80 (1986). https://doi.org/10.1007/BF00141686
Issue Date:
DOI: https://doi.org/10.1007/BF00141686