Symmetry extensions of “neutrality” I: Advantage to the condorcet loser

This is the first of three papers introducing a theory for positional voting methods that determines all possible election rankings and relationships that ever could occur with a profile over all possible subsets of candidates for any specified choices of positional voting methods. As such, these results extend to all positional voting systems what was previously possible only for the Borda Count and the plurality voting systems. In this first part certain mathematical symmetries based on neutrality are used 1) to generalize the basic properties that cause the desired features of the Borda Count and 2) to describe classes of positional voting methods with new types of election relationships among the election outcomes. Some of these relationships generalize the well-known results about the positioning of a Condorcet winner/loser within a Borda ranking, but now it is possible for the Condorcet loser, rather than the winner, to have the advantage to win certain positional elections. Included among the results are axiomatic characterizations of many positional voting methods.This research was supported in part by NSF Grant IRI-9103180.     

Millions of election outcomes from a single profile

By changing the choice of a positional voting method, different election rankings can result from a fixed profile. A geometric theory is developed to explain why this occurs, to completely characterize all possible sets of rankings that can arise in this manner, to determine the number of rankings and other properties of these sets of rankings, to design profiles that cause the different conclusions, to develop elementary tools to analyze actual data, and to compare new types of social choice solutions that are based on the set of rankings admitted by a profile. A secondary theme is to indicate how results for voting theory can be obtained with (relative) ease when they are analyzed with a geometric approach.This research was supported by NSF Grant IRI-9103180.     

The Borda dictionary

Forn candidates, a profile of voters defines a unique Borda election ranking for each of the 2 ⁿ – (n + 1) subsets of two or more candidates. The Borda Dictionary is the set of all of these election listings that occur for any choice of a profile. As such, the dictionary contains all positive features, all flaws, and all paradoxes that can occur with single profile, sincere Borda elections. After the Borda Dictionary is characterized, it is used to show in what ways the Borda Count (BC) is an improvement over other positional voting methods and to derive several new BC properties. These properties include several new characterizations of the BC expressed in terms of axiomatic representations of social choice functions, as well as showing, for example, that the BC ranking ofn candidates can be uniquely determined by the BC rankings of all sets ofk <n candidates for any choice ofk between 2 andn.This research was supported, in part, by NSF grants IRI8415348, IRI-8803505 and a Fellowship from the Guggenheim Memorial Foundation.     

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     

Exploiting polyhedral symmetries in social choice

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet’s paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.     

Symmetry extensions of “neutrality”

For n3 candidates, a system voting vector W ⁿ specifies the positional voting method assigned to each of the 2 ⁿ –(n+1) subsets of two or more candidates. While most system voting vectors need not admit any relationships among the election rankings; the ones that do are characterized here. The characterization is based on a particular geometric structure (an algebraic variety) that is described in detail and then used to define a partial ordering among system voting vectors. The impact of the partial ordering is that if W ⁿ ₁ W ⁿ ₂, then W ⁿ ₂ admits more kinds of (single profile) voting paradoxes than W ⁿ ₁. Therefore the partial ordering provides a powerful, computationally feasible way to compare system voting vectors. The basic ideas are illustrated with examples that completely describe the partial ordering for n=3 and n=4 candidates.This reearch was supported in part by NSF Grant IRI-9103180.     

On removing Condorcet effects from pairwise election tallies

Condorcet profiles are responsible for a number of well known preference aggregation paradoxes. It is reasonable to claim that these profiles represent complete ties between the candidates and should therefore be excluded to determine election outcomes. Established profile decomposition techniques are of limited usefulness in extracting and removing Condorcet effects, because of the computational complexity involved, even if complete knowledge of voters’ rankings of all candidates are available. The paper discusses an easily implementable method of removing or reducing Condorcet effects from pairwise scores. Pairwise scores (and not complete knowledge of voters’ rankings) are often the only available data based on which an election winner has to be determined.     

Different least square values,different rankings

The semivalues (as well as the least square values) propose different linear solutions for cooperative games with transferable utility. As a byproduct, they also induce a ranking of the players. So far, no systematic analysis has studied to which extent these rankings could vary for different semivalues. The aim of this paper is to compare the rankings given by different semivalues or least square values for several classes of games. Our main result states that there exist games, possibly superadditive or convex, such that the rankings of the players given by several semivalues are completely different. These results are similar to the ones D. Saari discovered in voting theory: There exist profiles of preferences such that there is no relationship among the rankings of the candidates given by different voting rules. Received: 5 November 2000/Accepted: 12 February 2001     

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.     

Community Structure and Voting in the South

This article combines census and survey data to assess the effectsof community characteristics, black concentration and urbanism,on the decision processes of southern white voters. Examiningvoting for Wallace in the 1968 election, the effects of blackconcentration on voting, attitudes, issue salience, and perceptionsof candidates are found to be generally greater in rural thanin urban areas.     

中国城市的社会阶层与基层人大选举

与强调用民主价值观、政治效能感解释中国基民人大选举中人们投票行为的思路不同,作者将阶层政治论的基本逻辑用于解释投票行为。该理论的理性选择解释把阶层利益作为核心阐明变量,而阶层政治的社会心理解释则强调阶层认同的不可或缺性。运用逻辑斯蒂回归及检验模型间系数差异的自助法对CGSS2006数据的分析结果,支持了基于这两种解释的研究假设。公职新中产、市场新中产、小业主与自雇者均比工人更有可能参加投票;而这又由其收入更高、更认同中产阶层来阐明;收入越高越认同中产阶层;收入越高、越认同中产阶层就越有可能参加投票。归纳这些研究发现,并进一步整合理性选择与社会心理解释,作者认为阶层利益不但是联系阶层地位与投票行为的机制,还通过阶层认同阐明阶层地位与投票行为间的因果关联;阶层政治论是分析当前中国民众政治参与的一种有效理论工具。     

When winning is the only thing: pure strategy Nash equilibria in a three-candidate spatial voting model

It is well known that there are no pure strategy Nash equilibria (PSNE) in the standard three-candidate spatial voting model when candidates maximize their share of the vote. When all that matters to the candidates is winning the election, however, we show that PSNE do exist. We provide a complete characterization of such equilibria and then extend our results to elections with an arbitrary number of candidates.     

Out of Step? Voters and Social Issues in U.S. Presidential Elections

Recent political commentary has argued that the Republican Party is “out of step” with voters on social issues, costing Republicans the 2012 presidential election. This dramatic claim is deserving of scrutiny in its own right and also for the way it offers new perspective on long‐standing controversies concerning the role of social issues in U.S. national elections. We present results that seek to advance established scholarship on electoral politics as well as journalistic claims concerning the rising importance of social issues for elections. Using data from the American National Election Study, we find that social issues mattered to voters in presidential elections from 1992 through 2012. The influence of social issues on voter choice rivals those of attitudes toward defense spending and government provision. We find further evidence that liberalizing trends in social issue opinion consistently benefited Democratic candidates in presidential elections. We consider the relevance of these results for scholarship on voter choice and elections, noting further implications for commentary on the 2012 presidential election.     

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 nail-biting election

In the first competitive election for President of the Social Choice and Welfare Society, the (official) approval-voting winner differed from the (hypothetical) Borda count winner, who was also the Condorcet winner. But because the election was essentially a toss-up, it is impossible to say who should have won. The election for Council was more true to form of other professional-society elections, with the winners identical, and even their rankings almost duplicative, under both voting systems. Received: 11 April 2000/Accepted: 26 March 2001     

A geometric examination of Kemeny's rule

By using geometry, a fairly complete analysis of Kemeny's rule (KR) is obtained. It is shown that the Borda Count (BC) always ranks the KR winner above the KR loser, and, conversely, KR always ranks the BC winner above the BC loser. Such KR relationships fail to hold for other positional methods. The geometric reasons why KR enjoys remarkably consistent election rankings as candidates are added or dropped are explained. The power of this KR consistency is demonstrated by comparing KR and BC outcomes. But KR's consistency carries a heavy cost; it requires KR to partially dismiss the crucial “individual rationality of voters” assumption. Received: 5 February 1998/Accepted: 26 May 1999     

Pre-election polling and third party candidates

I analyze voters’ incentives in responding to pre-election polls with a third party candidate. Third party supporters normally have an incentive to vote strategically in the election by voting for one of the major candidates. But these voters would vote third party if the third party candidate is doing surprisingly well in the polls. Because voters are more likely to vote third party if the third party candidate is doing well in polls, voters who like the third party candidate best have an incentive to claim they will vote third party in the polls so that more voters will ultimately vote third party in the election. The differing incentives faced during polls and elections accounts for why third party candidates do better in polls than in elections.     

Market arbitrage, social choice and the core

 This paper establishes a clear connection between equilibrium theory, game theory and social choice theory by showing that, for a well defined social choice problem, a condition which is necessary and sufficient to solve this problem – limited arbitrage – is the same as the condition which is necessary and sufficient to establish the existence of an equilibrium and the core. The connection is strengthened by establishing that a market allocation, which is in the core, can always be realized as a social allocation, i.e. an allocation which is optimal according to an ordering chosen by a social choice rule. Limited arbitrage characterizes those economies without Condorcet triples, and those for which Arrow’s paradox can be resolved on choices of large utility values. Received: 30 December 1994/Accepted: 12 August 1996     

Religion and Voting: A Study of the 1968 Canadian Federal Election in Ontario

On a cherchC A ddceler le genre de liaisons qui existaient entre certaines dimensions de la ferveur religieuse et les prdfdrences dans le vote a l'aide de donndes recueillies sur questionnaire dam la Province de l'Ontario au moment de 1'Clection fCdCrale canadienne de 1968. Les dimensions de la ferveur religieuse sont les suivantes: engagement social dans la communautd religieuse, orthodoxie doctrinale, I'intensitC de la pratique religieuse, les attitudes vis-his les autres groupes religieux et la question des Ccoles dpardes, et l'affiliation religieuse des candidats. Les mesures de l'engagement social sont trks dtroitement reliees A celles des preferences dans le vote tandis que la religion des interrogks ne produit aucun resultat tangible. La liaison entre la religion et le vote est de meme nature et intensite tant au niveau provincial que federal.
Propositions on the association of a number of dimensions of religiosity with voting preferences were tested by survey data collected in Ontario at the time of the 1968 Canadian federal election. The religiosity variables include: social involvement in the religious community, doctrinal orthodoxy, devotionalism, attitudes to other religious groups and to the separate schools issue, and the religion of the candidates. Measures of social involvement had the strongest association with voting preferences while the religion of the candidate had no significant effect. The association between religion and voting was of the same kind and strength at the provincial level as at the federal.     

Simple plurality versus plurality runoff with privately informed voters

This paper compares two voting methods commonly used in presidential elections: simple plurality voting and plurality runoff. In a situation in which a group of voters have common interests but do not agree on which candidate to support due to private information, information aggregation requires them to split their support between their favorite candidates. However, if a group of voters split their support, they increase the probability that the winner of the election is not one of their favorite candidates. In a model with three candidates, due to this tension between information aggregation and the need for coordination, plurality runoff leads to higher expected utility for the majority than simple plurality voting if the information held by voters about the candidates is not very accurate. Received: 12 September 2000/Accepted: 8 November 2001