The majority rule with a chairman

For the case of two alternatives and a given finite set of at least three individuals, seven axioms are shown to characterize the rules that are either the relative majority rule or the relative majority in which a given individual, the chairman, can always break ties. An axiomatization of the relative majority rules with a chairman is suggested that holds for an even number of individuals and that, for an odd number of individuals, characterizes the rules that are either the relative majority rule or a relative majority rule with a chairman.     

Scoring rules for judgment aggregation

This paper introduces a new class of judgment aggregation rules, to be called ‘scoring rules’ after their famous counterparts in preference aggregation theory. A scoring rule generates the collective judgment set which reaches the highest total ‘score’ across the individuals, subject to the judgment set having to be rational. Depending on how we define ‘scores’, we obtain several (old and new) solutions to the judgment aggregation problem, such as distance-based aggregation, premise- and conclusion-based aggregation, truth-tracking rules, and a generalization of the Borda rule to judgment aggregation theory. Scoring rules are shown to generalize the classical scoring rules of preference aggregation theory.     

Social compromise and social metrics

In the context of constitutional design, a committee of citizens' representatives proposes (social choice) axioms. The outcome should not necessarily be the aggregation rule, or rules, satisfying each of the axioms. Alternative procedures are recommended for aggregating a set of axioms into a single summary axiom. The summary axiom generates a single social decision function when a social metric is applied.     

Borda’s Paradox with weighted scoring rules

Representations are obtained for the probabilities that a Strict Borda Paradox and a Strong Borda Paradox are observed for large electorates with three candidates under the standard assumptions of Impartial Culture and Impartial Anonymous Culture. These representations are obtained for general weighted scoring rules (WSRs), and the probabilities are found to be maximized for voting rules like plurality rule and negative plurality rule. It is found that these paradox probabilities are not reduced for every scoring rule with the introduction of some degree of dependence among voters’ preferences with IAC. It is concluded that actual observances of a Strict Borda Paradox should be extremely rare, and that while observances of a Strong Borda Paradox should also be rare, they might occasionally be witnessed.     

Pairwise Strategy-Proofness and Self-Enforcing Manipulation

“Strategy-proofness” is one of the axioms that are most frequently used in the recent literature on social choice theory. It requires that by misrepresenting his preferences, no agent can manipulate the outcome of the social choice rule in his favor. The stronger requirement of “group strategy-proofness” is also often employed to obtain clear characterization results of social choice rules. Group strategy-proofness requires that no group of agents can manipulate the outcome in their favors. In this paper, we advocate “effective pairwise strategy-proofness.” It is the requirement that the social choice rule should be immune to unilateral manipulation and “self-enforcing” pairwise manipulation in the sense that no agent of a pair has the incentive to betray his partner. We apply the axiom of effective pairwise strategy-proofness to three types of economies: public good economy, pure exchange economy, and allotment economy. Although effective pairwise strategy-proofness is seemingly a much weaker axiom than group strategy-proofness, effective pairwise strategy-proofness characterizes social choice rules that are analyzed by using different axioms in the literature.     

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.     

Equality of Opportunity versus Equality of Opportunity Sets

We characterize two different approaches to the idea of equality of opportunity. Roemer’s social ordering is motivated by a concern to compensate for the effects of certain (non-responsibility) factors on outcomes. Van de gaer’s social ordering is concerned with the equalization of the opportunity sets to which people have access. We show how different invariance axioms open the possibility to go beyond the simple additive specification implied by both rules. This offers scope for a broader interpretation of responsibility-sensitive egalitarianism.     

Voting rules as statistical estimators

We adopt an ‘epistemic’ interpretation of social decisions: there is an objectively correct choice, each voter receives a ‘noisy signal’ of the correct choice, and the social objective is to aggregate these signals to make the best possible guess about the correct choice. One epistemic method is to fix a probability model and compute the maximum likelihood estimator (MLE), maximum a posteriori (MAP) estimator or expected utility maximizer (EUM), given the data provided by the voters. We first show that an abstract voting rule can be interpreted as MLE or MAP if and only if it is a scoring rule. We then specialize to the case of distance-based voting rules, in particular, the use of the median rule in judgement aggregation. Finally, we show how several common ‘quasiutilitarian’ voting rules can be interpreted as EUM.     

Subgame perfect implementation of voting rules via randomized mechanisms

We characterize completely ordinal and onto choice rules that are subgame perfect of Nash equilibrium (SPE) implementable via randomized mechanisms under strict preferences. The characterization is very operationalizable, and allows us to analyse SPE implementability of voting rules. We show that no scoring rule is SPE implementable. However, the top-cycle and the uncovered correspondences as well as plurality with runoff and any strongly Condorcet consistent voting rule can be SPE implemented. Therefore our results are favourable to majority based voting rules over scoring rules. Nevertheless, we show that many interesting Condorcet consistent but not strongly Condorcet consistent rules, such as the Copeland rule, the Kramer rule and the Simpson rule, cannot be SPE implemented.     

On the selection of the same winner by all scoring rules

Different scoring rules can result in the selection of any of the k competing candidates, given the same preference profile, (Saari DG 2001, Chaotic elections! A mathematician looks at voting. American Mathematical Society, Providence, R.I.). It is also possible that a candidate, and even a Condorcet winning candidate, cannot be selected by any scoring rule, (Saari DG 2000 Econ Theory 15:55–101). These findings are balanced by Saari’s result (Saari DG 1992 Soc Choice Welf 9(4):277–306) that specifies the necessary and sufficient condition for the selection of the same candidate by all scoring rules. This condition is, however, indirect. We provide a sufficient condition that is stated directly in terms of the preference profile; therefore, its testability does not require the verdict of any voting rule.     

Characteristic properties of list proportional representation systems

In this paper, three characterizations are given of a rule that models list systems of proportional representation the plurality ranking rule. It is shown that a social preference rule is the plurality ranking rule if and only if it satisfies three independent conditions: consistency, faithfulness, and first score cancellation. It is also shown that first score cancellation is implied by neutrality, anonymity, and topsonlyness. This means a second characterization is found, containing deeper axioms than the previous one. A third characterization contains the notion of top monotonicity. In order to motivate topsonlyness, we show that a scoring seat allocation rule is proof against party fragmentation if and only if it is topsonly. Various other properties of the plurality ranking rule are related to its characterist properties.     

Minimal monotonic extensions of scoring rules

Noting the existence of social choice problems over which no scoring rule is Maskin monotonic, we characterize minimal monotonic extensions of scoring rules. We show that the minimal monotonic extension of any scoring rule has a lower and upper bound, which can be expressed in terms of alternatives with scores exceeding a certain critical score. In fact, the minimal monotonic extension of a scoring rule coincides with its lower bound if and only if the scoring rule satisfies a certain weak monotonicity condition (such as the Borda and antiplurality rule). On the other hand, the minimal monotonic extension of a scoring rule approaches its upper bound as its degree of violating weak monotonicity increases, an extreme case of which is the plurality rule with a minimal monotonic extension reaching its upper bound.

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On the characterization of liberalism by Samet and Schmeidler

In the model of group identification, Samet and Schmeidler (J Econ Theory 110:213–233, 2003) provide two axiomatic characterizations of the“liberal” decision rule (a person is socially qualified as a member of a collective if and only if he qualifies himself). They impose standard monotonicity, non-degeneracy, and independence axioms, together with either exclusive self-determination (opinions by disqualified persons about qualified persons should not matter) or affirmative self-determination (social decision on who are qualified should coincide with social decision on who should be the qualifiers). Dropping monotonicity (and also non-degeneracy in a result) and considering a more general domain to allow neutral opinions, we characterize a larger family of “self-dependent” rules that share the important feature of liberalism that qualification of i depends only on i’s own opinion about himself or herself. Samet and Schmeidler’s results can be obtained with a weaker set of axioms and with a more general domain condition.     

A note on the redundancy of an axiom in the Pattanaik-Peleg characterization of the lexicographic maximin extension

P. K. Pattanaik and B. Peleg have provided an axiomatic characterization of the lexicographic maximin extension of an ordering over a finite set to an ordering over the set of all non-empty subsets of that set. This note shows that one of their axioms, namely the Union axiom, is redundant, thus obtaining a more economical characterization composed of the remaining four axioms (which are independent by virtue of the examples given by Pattanaik and Peleg).     

The inverse plurality rule—an axiomatization

Under the ‘inverse plurality rule’, voters specify only their least preferred alternative. Our first result establishes that this rule is the only scoring rule that satisfies the minimal veto condition (MV). We then prove that the inverse plurality rule is characterized by MV and the four well known conditions that characterize scoring rules; namely, Anonymity (A), Neutrality (N), Reinforcement (RE) and Continuity (CO). Our new characterization result is related to the characterizations of approval voting and of the widely used plurality rule. We finally show how the axiomatization of the inverse plurality rule can be extended to the axiomatization of elementary scoring rules (vote for t-alternatives scoring rules). We are indebted to two anonymous referees for their most useful comments.

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Quasi-transitive and Suzumura consistent relations

We examine properties of binary relations that complement quasi-transitivity and Suzumura consistency in the sense that they, in conjunction with the original axiom(s), are equivalent to transitivity. In general, the conjunction of quasi-transitivity and Suzumura consistency is weaker than transitivity but in the case of collective choice rules that satisfy further properties, this conjunction implies transitivity of the social relation. We prove this observation by characterizing the Pareto rule as the only collective choice rule such that collective preference relations are quasi-transitive and Suzumura consistent, and standard social choice axioms are satisfied.     

Scoring rule voting games and dominance solvability

This article studies the dominance solvability (by iterated deletion of weakly dominated strategies) of general scoring rule voting games when there are three alternatives. The scoring rules we study include Plurality rule, Approval voting, Borda rule, and Relative Utilitarianism. We provide sufficient conditions for dominance solvability of general scoring rule voting games. The sufficient conditions that we provide for dominance solvability are in terms of one statistic of the game: sufficient agreement on the best alternative or on the worst alternative. We also show that the solutions coincide with the set of Condorcet Winners whenever the sufficient conditions for dominance solvability are satisfied. Approval Voting performs the best in terms of our criteria.     

On the likelihood of Condorcet's profiles

Consider a group of individuals who have to collectively choose an outcome from a finite set of feasible alternatives. A scoring or positional rule is an aggregation procedure where each voter awards a given number of points, w _j, to the alternative she ranks in j ^th position in her preference ordering; The outcome chosen is then the alternative that receives the highest number of points. A Condorcet or majority winner is a candidate who obtains more votes than her opponents in any pairwise comparison. Condorcet [4] showed that all positional rules fail to satisfy the majority criterion. Furthermore, he supplied a famous example where all the positional rules select simultaneously the same winner while the majority rule picks another one. Let P ^* be the probability of such events in three-candidate elections. We apply the techniques of Merlin et al. [17] to evaluate P ^* for a large population under the Impartial Culture condition. With these assumptions, such a paradox occurs in 1.808% of the cases. Received: 30 April 1999/Accepted: 14 September 2000     

On the manipulability of approval voting and related scoring rules

We characterize all preference profiles at which the approval (voting) rule is manipulable, under three extensions of preferences to sets of candidates: by comparison of worst candidates, best candidates, or by comparison based on stochastic dominance. We perform a similar exercise for k-approval rules, where voters approve of a fixed number k of candidates. These results can be used to compare (k-)approval rules with respect to their manipulability. Analytical results are obtained for the case of two voters, specifically, the values of k for which the k-approval rule is minimally manipulable—has the smallest number of manipulable preference profiles—under the various preference extensions are determined. For the number of voters going to infinity, an asymptotic result is that the k-approval rule with k around half the number of candidates is minimally manipulable among all scoring rules. Further results are obtained by simulation and indicate that k-approval rules may improve on the approval rule as far as manipulability is concerned.     

Manipulation via Endowments in Exchange Markets with Indivisible Goods

We consider exchange markets with heterogeneous indivisible goods. We are interested in exchange rules that are efficient and immune to manipulations via endowments (either with respect to hiding or destroying part of the endowment or transferring part of the endowment to another trader). We consider three manipulability axioms: hiding-proofness, destruction-proofness, and transfer-proofness. We prove that no rule satisfying efficiency and hiding-proofness (which together imply individual rationality) exists. For two agents with separable and responsive preferences, we show that efficient, individually rational, and destruction-proof rules exist. However, for some profiles of separable preferences, no rule is efficient, individually rational, and destruction-proof. In the case of transfer-proofness the compatibility with efficiency and individual rationality for the two-agent case extends to the unrestricted domain. If there are more than two agents, for some profiles of separable preferences, no rule is efficient, individually rational, and transfer-proof.