Variable-population voting rules

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

[1]  Yongsheng Xu,et al.  A general scoring rule , 2012, Math. Soc. Sci..

[2]  Amrita Dhillon.  Extended Pareto rules and relative utilitarianism , 1998 .

[3]  I. Gilboa,et al.  Inductive Inference: An Axiomatic Approach , 2001 .

[4]  D. G. Saari,et al.  Consistency of decision processes , 1990 .

[5]  S. Ching.  A Simple Characterization of Plurality Rule , 1996 .

[6]  D. Black.  The theory of committees and elections , 1959 .

[7]  M. Pivato.  Additive representation of separable preferences over infinite products , 2014 .

[8]  Duane A. Cooper.  The Potential of Cumulative Voting To Yield Fair Representation , 2007 .

[9]  Peter C. Fishburn,et al.  Axioms for approval voting: Direct proof , 1978 .

[10]  H. Young,et al.  A Consistent Extension of Condorcet’s Election Principle , 1978 .

[11]  H. P. Young,et al.  A Note on Preference Aggregation , 1974 .

[12]  H. Young.  Social Choice Scoring Functions , 1975 .

[13]  D. J Hartfiel,et al.  A characterization result for the plurality rule , 1978 .

[14]  Roger B. Myerson,et al.  Axiomatic derivation of scoring rules without the ordering assumption , 1993 .

[15]  Carlos Alós-Ferrer,et al.  A Simple Characterization of Approval Voting , 2006, Soc. Choice Welf..

[16]  H. P. Young,et al.  An axiomatization of Borda's rule , 1974 .

[17]  J. H. Smith.  AGGREGATION OF PREFERENCES WITH VARIABLE ELECTORATE , 1973 .

[18]  Ariel Rubinstein,et al.  A further characterization of Borda ranking method , 1981 .

[19]  Donald G. Saari.  Systematic analysis of multiple voting rules , 2010, Soc. Choice Welf..

[20]  Chun-Hsien Yeh.  An efficiency characterization of plurality rule in collective choice problems , 2008 .

[21]  Marcus Pivato,et al.  Voting rules as statistical estimators , 2013, Soc. Choice Welf..

[22]  J. G. Wendel,et al.  ORDERED VECTOR SPACES , 1952 .

[23]  William S. Zwicker.  Consistency without neutrality in voting rules: When is a vote an average? , 2008, Math. Comput. Model..