Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

“Subset voting” denotes a choice situation where one fixed set of choice alternatives (candidates, products) is offered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, “approval voting“ from political science, the “weak utility model” from mathematical psychology, and “social welfare orderings” from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sufficient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988). Received: 18 August 1995 / Accepted: 13 February 1997     

On the (Sample) Condorcet Efficiency of Majority Rule: An alternative view of majority cycles and social homogeneity

The Condorcet efficiency of a social choice procedure is usually defined as the probability that this procedure coincides with the majority winner (or majority ordering) in random samples, given a majority winner exists (or given the majority ordering is transitive). Consequently, it is in effect a conditional probability that two sample statistics coincide, given certain side conditions. We raise a different issue of Condorcet efficiencies: What is the probability that a social choice procedure applied to a sample matches with the majority preferences of the population from which the sample was drawn? We investigate the canonical case where the sample statistic is itself also majority rule and the samples are drawn from real world distributions gathered from national election surveys in Germany, France, and the United States. We relate the results to the existing literature on majority cycles and social homogeneity. We find that these samples rarely display majority cycles, whereas the probability that a sample misrepresents the majority preferences of the underlying population varies dramatically and always exceeds the probability that the sample displays cyclic majority preferences. Social homogeneity plays a fundamental role in the type of Condorcet efficiency investigated here.

     

On the probabilities of correct or incorrect majority preference relations

While majority cycles may pose a threat to democratic decision making, actual decisions based inadvertently upon an incorrect majority preference relation may be far more expensive to society. We study majority rule both in a statistical sampling and a Bayesian inference framework. Based on any given paired comparison probabilities or ranking probabilities in a population (i.e., culture) of reference, we derive upper and lower bounds on the probability of a correct or incorrect majority social welfare relation in a random sample (with replacement). We also present upper and lower bounds on the probabilities of majority preference relations in the population given a sample, using Bayesian updating. These bounds permit to map quite precisely the entire picture of possible majority preference relations as well as their probabilities. We illustrate our results using survey data. Received: 13 November 2000/Accepted: 19 March 2002 This collaborative work was carried out while Regenwetter was a faculty member at the Fuqua School of Business, Duke University. We thank Fuqua for sponsoring our collaboration and the National Science Foundation for grant SBR-97-30076 to Michel Regenwetter. We are indebted to the editor and the referees, as well as to Jim Adams, Bob Clemen, Bernie Grofman, Bob Nau, Saša Pekeč, Jim Smith and Bob Winkler for helpful comments and suggestions.     

The impartial culture maximizes the probability of majority cycles

Many papers have studied the probability of majority cycles, also called the Condorcet paradox, using the impartial culture or related distributional assumptions. While it is widely acknowledged that the impartial culture is unrealistic, conclusions drawn from the impartial culture are nevertheless still widely advertised and reproduced in textbooks. We demonstrate that the impartial culture is the worst case scenario among a very broad range of possible voter preference distributions. More specifically, any deviation from the impartial culture over linear orders reduces the probability of majority cycles in infinite samples unless the culture from which we sample is itself inherently intransitive. We prove this statement for the case of three candidates and we provide arguments for the conjecture that it extends to any number of candidates.All three authors thank the Fuqua School of Business for supporting their research collaboration. Regenwetter and Grofman gratefully acknowledge the precious support of the National Science Foundation through grant #SBR-9730076 on Probabilistic Models of Social Choice (Methodology, Measurement and Statistics program). We are grateful to the referees and we thank Saa Peke for critical comments on an earlier draft. Grofman thanks Scott L. Feld for numerous reminders about the implausibility of the impartial culture assumption which helped lead to this paper.     

A general concept of scoring rules: general definitions,statistical inference,and empirical illustrations

We develop the general conceptual, mathematical and statistical foundations of behavioral social choice for scoring rules. Traditional scoring rules are difficult to assess empirically because one rarely observes the deterministic complete linear orders that they require as input. We provide a general concept of scoring rules in terms of a broad range of mathematical representations of preference or utility, namely arbitrary finite binary relations, probability distributions over such relations, real valued multi-criteria utility vectors and real valued random utility representations. We extend Regenwetter et al.’s (Behavioral social choice. Cambridge University Press, Cambridge, 2006) statistical framework to a more general setting. We illustrate the general modeling and statistical tools by applying them to four well known sets of survey data. We illustrate two potential problems that have previously received little attention and that deserve systematic study in the future: (1) Scoring rule outcomes can suffer from model dependence in that the social welfare functions computed from ballot, survey, or hypothetical data may depend on implicit or explicit modeling assumptions. (2) Scoring rule outcomes may suffer from low statistical confidence in that the correct assessment of social orders from empirical data can be far from certain. We also illustrate the empirical congruence among conceptually competing social choice methods.