Variable-population voting rules

Pivato, Marcus (2011): Variable-population voting rules.

Abstract

Let X be a set of social alternatives, and let V be a set of `votes' or `signals'. (We do not assume any structure on X or V). A `variable population voting rule' F takes any number of anonymous votes drawn from V as input, and produces a nonempty subset of X as output. The rule F satisfies `reinforcement' if, whenever two disjoint sets of voters independently select some subset Y of X, the union of these two sets will also select Y. We show that F satisfies reinforcement if and only if F is a `balance rule'. If F satisfies a form of neutrality, then F is satisfies reinforcement if and only if F is a scoring rule (with scores taking values in an abstract linearly ordered abelian group R); this generalizes a result of Myerson (1995). We also discuss the sense in which the balance or scoring representation of F is unique. Finally, we provide a characterization of two scoring rules: `formally utilitarian' voting and `range voting'. a

• Item Type: MPRA Paper
• Language: English
• Keywords: reinforcement; scoring rule; balance rule; linearly ordered abelian group; formal utilitarian; range voting
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations

References

Alos-Ferrer, C., 2006. A simple characterization of approval voting. Soc. Choice Welf. 27 (3), 621-625.

Brams, S. J., Fishburn, P. C., 1983. Approval voting. Birkhauser Boston, Mass.

Ching, S., 1996. A simple characterization of plurality rule. J. Econom. Theory 71 (1), 298-302.

Dhillon, A., 1998. Extended Pareto rules and relative utilitarianism. Soc. Choice Welf. 15 (4), 521-542.

Dhillon, A., Mertens, J.-F., 1999. Relative utilitarianism. Econometrica 67 (3), 471-498.

Dummit, D. S., Foote, R. M., 2004. Abstract algebra, 3rd Edition. John Wiley & Sons Inc., Hoboken, NJ.

Fishburn, P. C., 1978. Axioms for approval voting: direct proof. J. Econom. Theory 19 (1), 180-185.

Gaertner, W., Xu, Y., 2011. A general scoring rule. (preprint).

Hausner, M., Wendel, J. G., 1952. Ordered vector spaces. Proc. Amer. Math. Soc. 3, 977-982.

Morkelyunas, A., 1981. On the choice from collections of sets. Mat. Metody v Sotsial. Nauk.---Trudy Sem. Protsessy Optimal. Upravleniya (14), 33-49, 93.

Morkelyunas, A., 1982. Two choice rules similar to the plurality and Borda's rule. Mat. Metody v Sotsial. Nauk. (15), 27-36.

Myerson, R. B., 1995. Axiomatic derivation of scoring rules without the ordering assumption. Soc. Choice Welf. 12 (1), 59-74.

Nitzan, S., Rubinstein, A., 1981. A further characterization of Borda ranking method. Public Choice 36, 153-158.

Pivato, M., 2011. Additive representation of separable preferences over infinite products. (preprint).

Richelson, J. T., 1978. A characterization result for the plurality rule. J. Econom. Theory 19 (2), 548-550.

Smith, J. H., 1973. Aggregation of preferences with variable electorate. Econometrica 41, 1027-1041.

Smith,W. D., November 2000. Range voting, http://www.math.temple.edu/~wds/homepage/rangevote.pdf.

Yeh, C.-H., 2008. An efficiency characterization of plurality rule in collective choice problems. Econom. Theory 34 (3), 575-583.

Young, H. P., 1974a. An axiomatization of Borda’s rule. J. Econom. Theory 9 (1), 43-52.

Young, H. P., 1974b. A note on preference aggregation. Econometrica 42, 1129-1131.

Young, H. P., 1975. Social choice scoring functions. SIAM J. Appl. Math. 28, 824-838.

Young, H. P., Levenglick, A., 1978. A consistent extension of Condorcet's election principle. SIAM J. Appl. Math. 35 (2), 285-300.

Zwicker, W. S., 2008. Consistency without neutrality in voting rules: When is a vote an average? Math. Comput. Modelling 48 (9-10), 1357-1373.