The sensitivity of weight selection for scoring rules to profile proximity to single-peaked preferences

Niemi (Am Polit Sci Rev 63:488–497, 1969) proposed a simple measure of the cohesiveness of a group of n voters’ preferences that reflects the proximity of their preferences to single-peakedness. For three-candidate elections, this measure, k, reduces to the minimum number of voters who rank one of the candidates as being least preferred. The current study develops closed form representations for the conditional probability, PASW(n,IAC|k), that all weighted scoring rules will elect the Condorcet winner in an election, given a specified value of k. Results show a very strong relationship between PASW(n,IAC|k) and k, such that the determination of the voting rule to be used in an election becomes significantly less critical relative to the likelihood of electing the Condorcet winner as voters in a society have more structured preferences. As voters’ preferences become more unstructured as measured by their distance from single-peakedness, it becomes much more likely that different voting rules will select different winners.A preliminary version of this paper was presented at the European Public Choice Society Conference in Berlin, Germany, April 15–18, 2004.     

The replacement principle for the provision of multiple public goods on tree networks

This article considers the provision of two public goods on tree networks where each agent has a single-peaked preference. We show that if there are at least four agents, then no social choice rule exists that satisfies efficiency and replacement-domination. In fact, these properties are incompatible, even if agents’ preferences are restricted to a smaller domain of symmetric single-peaked preferences. However, for rules on an interval, we prove that Miyagawa’s (Soc Choice Welf 18:527–541, 2001) characterization that only the left-peaks rule and the right-peaks rule satisfy both of these properties also holds on the domain of symmetric single-peaked preferences. Moreover, if agents’ peak locations are restricted to either the nodes or the endpoints of trees, rules exist on a subclass of trees. We provide a characterization of a family of such rules for this tree subclass.     

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.     

Condorcet efficiencies under the maximal culture condition

The Condorcet winner in an election is a candidate that could defeat each other candidate in a series of pairwise majority rule elections. The Condorcet efficiency of a voting rule is the conditional probability that the voting rule will elect the Condorcet winner, given that such a winner exists. The study considers the Condorcet efficiency of basic voting rules under various assumptions about how voter preference rankings are obtained. Particular attention is given to situations in which the maximal culture condition is used as a basis for obtaining voter preferences. Received: 4 February 1998/Accepted: 13 April 1998     

A law of large numbers for weighted plurality

Consider an election between $k$ candidates in which each voter votes randomly (but not necessarily independently) for a single candidate, and suppose that there is a single candidate that every voter prefers (in the sense that each voter is more likely to vote for this special candidate than any other candidate). Suppose we have a voting rule that takes all of the votes and produces a single outcome and suppose that each individual voter has little effect on the outcome of the voting rule. If the voting rule is a weighted plurality, then we show that with high probability, the preferred candidate will win the election. Conversely, we show that this statement fails for all other reasonable voting rules. This result is an extension of one by Häggström, Kalai and Mossel, who proved the above in the case $k=2$ .     

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.     

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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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.     

Free triples,large indifference classes and the majority rule

We study classes of voting situations where agents may exhibit a systematic inability to distinguish between the elements of certain sets of alternatives. These sets of alternatives may differ from voter to voter, thus resulting in personalized families of preferences. We study the properties of the majority relation when defined on restricted domains that are the cartesian product of preference families, each one reflecting the corresponding agent’s objective indifferences, and otherwise allowing for all possible (strict) preference relations among alternatives. We present necessary and sufficient conditions on the preference domains of this type, guaranteeing that majority rule is quasi-transitive and thus the existence of Condorcet winners at any profile in the domain, and for any finite subset of alternatives. Finally, we compare our proposed restrictions with others in the literature, to conclude that they are independent of any previously discussed domain restriction.     

Policy convergence under approval and plurality voting: the role of policy commitment

Proponents of Approval Voting argue that this electoral rule leads to more centrist outcomes compared to Plurality Voting. This claim has been substantiated by scholarly work using spatial models of political competition. We revisit this issue in the context of a model of political competition in which (1) candidates are policy-motivated; (2) candidacy decisions are endogenous; and (3) candidates can credibly commit to implementing any policy. Under these assumptions we find the opposite to be true – Plurality Voting yields convergence to the median voter’s ideal policy but Approval Voting may not. We argue that this result is driven by the differential incentives for candidate entry under the two voting rules. Our results suggest that whether Approval Voting yields more centrist outcomes vis-á-vis Plurality Voting depends on the possibility of policy commitment on the part of the candidates. In an election held under PV each citizen is given one vote he can cast for one (and only one) candidate, and the candidate who gets the most votes wins the election.     

Changes that cause changes

Beyond determining whether procedures can be manipulated, the real goal for any analysis of “strategic behavior” is to identify all settings where and when this can be done, who can do it, and what they should do. By applying the geometric approach of Saari [7, 8] to the Kemeny's Rule (KR), we demonstrate how surprisingly simple this analysis can be, we identify all three candidate KR strategic behavior, and we show how an almost identical analysis answers most other multiple profile concerns (e.g., the abstention paradox and when voters just make errors). We also introduce new measures, which can be used with any procedure, to compare strategic and other behavior involving “changes.” These measures help to identify settings where it may be more important to worry about honest mistakes than strategic voting. Received: 16 April 1999/Accepted: 29 September 1999     

On the Average Minimum Size of a Manipulating Coalition

We study the asymptotic average minimum manipulating coalition size as a characteristic of quality of a voting rule and show its serious drawback. We suggest using the asymptotic average threshold coalition size instead. We prove that, in large electorates, the asymptotic average threshold coalition size is maximised among all scoring rules by the Borda rule when the number m of alternatives is 3 or 4, and by -approval voting when m ≥ 5.     

A note on the stability of Chen’s Lindahl mechanism

There are no general theoretical results on the stability of the Lindahl mechanism introduced by Chen (Econ Theory 19:773–790, 2002). We show that despite not fitting the requirements of the Milgrom and Roberts 1990 stability result for supermodular games, if the Chen mechanism induces a supermodular game, then the best reply map is a contraction. This gives us an easy to identify sufficient condition for dynamic stability of equilibrium.     

Generalized average rules as stable Nash mechanisms to implement generalized median rules

We consider a problem in which a policy is chosen from a one-dimensional set over which voters have single-peaked preferences. While Moulin (Public Choice 35:437–455, 1980) and others subsequent works have focused on strategy-proof rules, Renault and Trannoy (Mimeo 2011) and Renault and Trannoy (J Pub Econ Theory 7:169–199, 2005) have shown that the average rule implements a generalized median rule in Nash equilibria and provide an interpretation of the parameters in Moulin’s rule. In this article, we first extend their result by showing that a wide range of voting rules which includes the average rule can implement Moulin’s rule in Nash equilibria. Moreover, we show additionally that within this class, generalized average rules are Cournot stable. That is, from any strategy profile, any best response path must converge to a Nash equilibrium.     

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 equilibrium of voting games with abstention and several levels of approval

We study the core of “(j, k) simple games”, where voters choose one level of approval from among j possible levels, partitioning the society into j coalitions, and each possible partition facing k levels of approval in the output (Freixas and Zwicker in Soc Choice Welf 21:399–431, 2003). We consider the case of (j, 2) simple games, including voting games in which each voter may cast a “yes” or “no” vote, or abstain (j = 3). A necessary and sufficient condition for the non-emptiness of the core of such games is provided, with an important application to weighted symmetric (j, 2) simple games. These results generalize the literature, and provide a characterization of constitutions under which a society would allow a given number of candidates to compete for leadership without running the risk of political instability. We apply these results to well-known voting systems and social choice institutions including the relative majority rule, the two-thirds relative majority rule, the United States Senate, and the United Nations Security Council.     

Contraction consistent stochastic choice correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s α and β, and regularity. We identify a restriction on the domain of the stochastic choice correspondence (SCC), under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38:307–317, 1971) and Dasgupta and Pattanaik (Econ Theory 31:35–50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.     

Robustness against inefficient manipulation

This paper identifies a family of scoring rules that are robust against coalitional manipulations that result in inefficient outcomes. We discuss the robustness of a number of Condorcet consistent and “point runoff” voting rules against such inefficient manipulation and classify voting rules according to their potential vulnerability to inefficient manipulation.     

How to choose a non-controversial list with k names

Barberà and Coelho (WP 264, CREA-Barcelona Economics, 2007) documented six screening rules associated with the rule of k names that are used by diferent institutions around the world. Here, we study whether these screening rules satisfy stability. A set is said to be a weak Condorcet set à la Gehrlein (Math Soc Sci 10:199–209) if no candidate in this set can be defeated by any candidate from outside the set on the basis of simple majority rule. We say that a screening rule is stable if it always selects a weak Condorcet set whenever such set exists. We show that all of the six procedures which are used in reality do violate stability if the voters do not act strategically. We then show that there are screening rules which satisfy stability. Finally, we provide two results that can explain the widespread use of unstable screening rules.