Voting Rules
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When Ties are Possible: Weak Condorcet Winners and Arrovian Rationality
Mathematical Social Sciences 123: 128-136, 2023
Abstract. We use Ehrhart polynomials to estimate the likelihood of each three-candidate social ranking produced by pairwise majority rule assuming an even number of voters and the Impartial Anonymous Culture condition. We then calculate the probability the ranking is transitive and the probability of a weak Condorcet winner. Finally, we determine the weak Condorcet efficiency of various voting rules. We prove that Baldwin, Nanson, Copeland, and ranked pairs are weak Condorcet efficient, as is Borda unless there are no ties. Simulations show that among the rest, Dowdall is typically the most efficient rule for small voter groups and anti-plurality the least.
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Negative Voting Social Welfare Functions: A Characterization
Review of Economic Design 27: 125-132, 2023
Abstract. An Arrowian Social Welfare Function (SWF) represents a weak ordering of the alternative set derived entirely from individual preference orders. Negative voting considers only a voter's least preferred alternative. We introduce the majority loser opposition property which requires that any alternative ranked last by a majority of voters must be ranked last in the social ordering. We show a SWF satisfies anonymity, neutrality, reinforcement, continuity, and majority loser opposition if and only if it is equivalent to negative voting.
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Characterizing Plurality using the Majoritarian Condition: A New Proof and Implications for Other Scoring Rules
Public Choice 189: 335–346, 2021
Abstract. Plurality rule selects whichever alternative is most preferred by the greatest number of voters. The majoritarian principle states that if a simple majority of voters agree on the most preferred alternative, then it must be selected uniquely. Lepelley (1992) adopts a proof by contradiction approach to show that plurality is the only scoring rule satisfying the majoritarian principle. We make use of the relationship that majoritarianism implies faithfulness to present a new proof allowing us to derive limits on the size of the group for which a particular scoring rule will satisfy majoritarianism without restricting voter preferences. We then determine the limits for three specific faithful scoring rules where voters rank the alternatives: positive/negative voting, wherein one point is awarded to a voter's top preference and one point is subtracted from a voter's bottom preference; Borda, in which an equal increase in points is awarded to each successively higher rank; and Dowdall, for which rank points entail an harmonic sequence. Comparing these rules by the sizes of group and alternative set combinations for which they are majoritarian we find that Borda is dominated by positive/negative voting, and both are dominated by Dowdall. We also derive the relative point gaps between certain pairs of rankings beyond which a scoring rule will not be majoritarian for any group of more than two voters.
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The Probability of Violating Arrow's Conditions
European Journal of Political Economy 65, 101936, 2020
Abstract. Arrow's impossibility theorem shows that all preference aggregation rules (PARs) must violate a specific set of normative conditions (transitivity, Pareto, IIA, nondictatorship) over an unrestricted domain of preference profiles. However, the theorem does not address which PARs are more likely to violate those conditions across preference profiles. We compare the probabilities that ten PARs (plurality, anti-plurality, Hare, Nanson, Simpson-Kramer, Borda, Copeland, Dowdall, Coombs, and pairwise majority) violate Arrow's conditions. We prove that Borda, Copeland, Dowdall, and Coombs are less likely to violate IIA than the first five PARs, and they are less likely to violate Arrow's conditions jointly. In contrast, pairwise majority never violates IIA but can violate transitivity. Simulations with three alternatives reveal that among the PARs studied, pairwise majority is the most likely to satisfy Arrow's conditions jointly. Our results suggest pairwise majority represents a low probability of an intransitive social preference whereas the others represent much higher probabilities of an IIA violation. The trade-off may depend on the importance of avoiding cycles versus a rule's vulnerability to strategic voting.
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Symmetric Scoring Rules and a New Characterization of the Borda Count
Economic Inquiry 59: 287-299, 2020
Abstract. Young (1974) developed a classic axiomatization of the Borda rule. He proved it is the only voting rule satisfying the normative properties of decisiveness, neutrality, reinforcement, faithfulness and cancellation. Often overlooked is the uniqueness of Borda applies only to variable populations. We present a different set of properties which only Borda satisfies when both the set of voters and the set of alternatives can vary. It is also shown Borda is the only scoring rule which will satisfy all of the new properties when the number of voters stays fixed.
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A Note on Neutrality and Majority Rule with an Application to State Votes at the Constitutional Convention of 1787
Public Choice 167: 245-255, 2016
Abstract. Majority rule used in the legislative process has a bias toward the status quo. This implies that proposals are less likely to pass when the number of voters casting either "yes" or "no" votes sums to an even number rather than an odd number. The implication is weakly supported by examining state votes of 552 motions made at the 1787 Constitutional Convention. A difference is found in the expected direction but is not statistically significant at traditional levels.
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Properties and Paradoxes of Common Voting Rules
Handbook of Social Choice and Voting (Jac C. Heckelman and Nicholas R. Miller (eds.)), Edward Elgar Press, pp. 263-283, 2015
Abstract. This chapter describes common practical voting rules used to choose among multiple alternatives, including those that assign scores to each alternative based on a voter's ranking, those that require majority support and utilize run-offs if necessary, those that are based on pairwise majority rule, and those that involve proportional lotteries. This chapter compares these rules with respect to their normative properties and provides examples that illustrate seemingly 'paradoxical' violations of such properties by particular voting rules.
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Strategy Proof Scoring Rule Lotteries for Multiple Winners
Journal of Public Economic Theory 15: 108-123, 2013
Abstract. We develop a lottery procedure for selecting multiple winners that is strategy proof. The rule assigns points to each candidate based on any standard scoring rule method, and then uses one draw to select a single winning set of candidates in proportion to their collective score. In addition to being strategy proof, the lottery rule is also shown to have several other attractive normative properties. Violations of some other important normative properties are noted as well.
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On Voting by Proportional Lottery
Korean Journal of Public Choice 2: 1-11, 2007
Abstract. Lottery rules are rarely used beyond breaking a tie vote. Yet, proportional lotteries have several attractive features which are discussed here. First, lotteries break the tyranny of the majority. Second, any proportional lottery which uses only a single round of voting to determine the lottery weights followed by a single weighted draw will ensure sincere preference revelation by the voters. Third, lotteries respect many of the general axiomatic principals invoked for social choice rules, in a probabilistic sense.
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Winning Probabilities in a Pairwise Lottery System with Three Alternatives
Economic Theory 26: 607-617, 2005
Abstract. The pairwise lottery system is a multiple round voting procedure which chooses by lot a winner from a pair of alternatives to advance to the next round where in each round the odds of selection are based on each alternative's majority rule votes. We develop a framework for determining the asymptotic relative likelihood of the lottery selecting in the final round the Borda winner, Condorcet winner, and Condorcet loser for the three alternative case. We also show the procedure is equivalent to a Borda lottery when only a single round of voting is conducted. Finally, we present an alternative voting rule which yields the same winning probabilities as the pairwise lottery in the limiting case as the number of rounds of the pairwise lottery becomes large.
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Probabilistic Borda Rule Voting
Social Choice and Welfare 21: 455-468, 2003
Abstract. An alternative voting system, referred to as probabilistic Borda rule, is developed and analyzed. The winning alternative under this system is chosen by lottery where the weights are determined from each alternative's Borda score relative to all Borda points possible. Advantages of the lottery include the elimination of strategic voting on the set of alternatives under consideration and breaking the tyranny of majority coalitions. Disadvantages include an increased incentive for strategic introduction of new alternatives to alter the lottery weights, and the possible selection of a Condorect loser. Normative axiomatic properties of the system are also considered. It is shown this system satisfies the axiomatic properties of the standard Borda procedure in a probabilistic fashion.