Probabilistic Borda rule voting

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 alternatives 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 Condorcet 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.I thank Nic Tideman and Andrew Yates for helpful comments and especially appreciate detailed criticism from Keith Dougherty which greatly improved the exposition of the text. In addition, Don Saari, in his role as editor, suggested a number of clarifications and an important extension for which I am grateful. I accept full responsibility for any remaining shortcomings.     

A majority-rule characterization with multiple extensions

When May's necessary and sufficient conditions for majority rule as a binary voting rule are extended in a natural way to decisions over more than two options, the resulting conditions are consistent with the Borda and Black voting rules, but not with a variety of other voting rules for more than two options. This paper presents an alternative set of necessary and sufficient conditions for binary majority rule, which permits the plurality, Condorcet and (simplified) Dodgson rules, as well as the Borda and Black rules, but not the Copeland or Nanson rules, to be classified as extensions of binary majority rule to decisions over more than two options.I am indebted to Amartya Sen and anonymous referees for helpful suggestions     

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.     

A comparison of theoretical and empirical evaluations of the Borda Compromise

The Borda Compromise states that, if one has to choose among five popular voting rules that are not Condorcet consistent, one should always give preference to the Borda rule over the four other rules. We assess the theoretical as well as the empirical support for the Borda Compromise. We find that, despite considerable differences between the properties of the theoretical framework and the characteristics of two sets of observed ranking data, all three analyses provide considerable support for the Borda Compromise.     

Why the count de Borda cannot beat the Marquis de Condorcet

Although championed by the Marquis the Condorcet and many others, majority rule has often been rejected as indeterminate, incoherent, or implausible. Majority rule's arch competitor is the Borda count, proposed by the Count de Borda, and there has long been a dispute between the two approaches. In several publications, Donald Saari has recently presented what is arguably the most vigorous and systematic defense of Borda ever developed, a project Saari has supplemented with equally vigorous objections to majority rule. In this article I argue that both Saari's objections to majority rule and his positive case for the Borda count fail. I hold the view that defenders of Condorcet cannot muster arguments to convince supporters of Borda, and vice versa, but here I am only concerned to show that the Count de Borda cannot beat the Marquis de Condorcet. Saari's approach displays what I take to be widespread fallacies in reasoning about social choice worthy of closer analysis. This debate bears on important questions in the philosophy of social choice theory.     

First and second best voting rules in committees

A committee of people with common preferences but different abilities in identifying the best alternative (e.g., a jury) votes in order to decide between two alternatives. The first best voting rule is a weighted voting rule that takes the different individual competences into account, and is therefore not anonymous, i.e., the voters’ identities matter. Under this rule, it is rational for the committee members to vote according to their true opinions, or informatively. This is not necessarily true for an anonymous voting rule, under which members may have an incentive to vote non-informatively. Thus, strategic, sophisticated voters may vary their voting strategies according to the voting rule rather than naively voting informatively. This paper shows that the identity of the best anonymous and monotone (i.e., quota) voting rule does not depend on whether the committee members are strategic or naive or whether some are strategic and some are naive. One such rule, called the second best rule, affords the highest expected utility in all cases.

“Wasn’t he sweet?” said Yossarian. “Maybe they should give him three votes.” Joseph Heller, Catch-22

     

Some Further Results on the Manipulability of Social Choice Rules

A typology of electoral environments in which strategic manipulation of voting rules can occur is proposed. From this typology, six different concepts of equilibria (or quasi-equilibria) are defined, including the usual ones. Then, focusing on three alternative elections, we derive some analytical representations for the vulnerability of a large number of voting rules to strategic manipulation by considering successively each type of equilibrium. Our results show that the hierarchy of the rules changes when the voting framework is modified. However, some rules are clearly dominated and some others (among which the Borda and the Nanson rules) appear to be uniquely the best in some specific contexts.     

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.     

The traveling group problem

This paper combines social choice theory with mathematical optimization by applying various group decision concepts to a classical problem of combinatorial optimization, namely the famous traveling salesperson (salesman) problem. The aim of the latter is to find a tour through all vertices of a given graph along edges of minimal total cost. In this contribution we replace the measure of additive edge costs by the social acceptance of different edges and the resulting tours. In particular, for four different voting rules, the Borda rule, Approval voting, Plurality rule and Simple Majority rule, we will investigate the social acceptance of tours derived from global and local decisions. It will be shown that these two decision approaches can lead to widely varying results.     

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.     

SYMMETRIC SCORING RULES AND A NEW CHARACTERIZATION OF THE BORDA COUNT

Young developed a classic axiomatization of the Borda rule almost 50 years ago. He proved it is the only voting rule satisfying the normative properties of decisiveness, neutrality, reinforcement, faithfulness and cancellation. Often overlooked is that 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. (JEL D71, D02, H00)     

Women's agency in living apart together: constraint,strategy and vulnerability

Recent research suggests that women can use living apart together (LAT) for a reflexive and strategic undoing of the gendered norms of cohabitation. In this article we examine this assertion empirically, using a representative survey from Britain in 2011 and follow‐up interviews. First, we find little gender differentiation in practices, expectations, or attitudes about LAT, or reasons for LAT. This does not fit in with ideas of undoing gender. Secondly, in examining how women talk about LAT in relation to gender, we distinguish three groups of ‘constrained’, ‘strategic’ and ‘vulnerable’ female interviewees. All valued the extra space and time that LAT could bring, many welcomed some release from traditional divisions of labour, and some were glad to escape unpleasant situations created by partnership with men. However, for the constrained and vulnerable groups LAT was second best, and any relaxation of gendered norms was seen as incidental and inconsequential to their major aim, or ideal, of the ‘proper family’ with cohabitation and marriage. Rather, their agency in achieving this was limited by more powerful agents, or was a reaction to perceived vulnerability. While the strategic group showed more purposeful behaviour in avoiding male authority, agency remained relational and bonded. Overall we find that women, at least in Britain, seldom use LAT to purposefully or reflexively undo gender. Equally, LAT sometimes involves a reaffirmation of gendered norms. LAT is a multi‐faceted adaption to circumstances where new autonomies can at the same time incorporate old subordinations, and new arrangements can herald conventional family forms.     

The two-party system under alternative voting procedures

Countries that elect their policy-makers by means of Plurality Voting tend to have a two-party system. This observation can be explained by the strategic behavior of voters. This article derives two broad classes of voting procedures under which strategic voting behavior induces a two-party system under standard assumptions on voter preferences. One class consists of the voting procedures with unique top-score, i.e., under which a voter can cast a top-score vote for only one candidate (e.g., Plurality Voting, Borda Count). The other class consists of the voting procedures that permit truncated ballots, i.e., under which voters do not have to cast all their votes (e.g., Approval Voting). This analysis suggests that the key for strategic voting behavior to induce a two-party system is that voters can always cast a different score for the two candidates they rank first and second on their ballots.     

The original Borda count and partial voting

In a Borda count, bc, M. de Borda suggested the last preference cast should receive 1 point, the voter’s penultimate ranking should get 2 points, and so on. Today, however, points are often awarded to (first, second,..., last) preferences cast as per (n, n?1, ..., 1) or more frequently, (n ?1, n?2,..., 0). If partial voting is allowed, and if a first preference is to be given n or n ? 1 points regardless of how many preferences the voter casts, he/she will be incentivised to rank only one option/candidate. If everyone acts in this way, the bc metamorphoses into a plurality vote... which de Borda criticized at length. If all the voters submit full ballots, the outcome—social choice or ranking—will be the same under any of the above three counting procedures. In the event of one or more persons submitting a partial vote, however, outcomes may vary considerably. This preliminary paper suggests research should consider partial voting. The author examines the consequences of the various rules so far advocated and then purports that M. de Borda, in using his formula, was perhaps more astute than the science has hitherto recognised.     

Ramon Llull: from ‘Ars electionis’ to social choice theory

Ramon Llull (Majorca c.1232–1316) is one of the earliest founding fathers of voting theory and social choice theory. The present article places Llull’s contributions and discussion in the historical context of elections in the medieval Church and the emergence of majority rule as a new general principle for making enforceable collective decisions in replacement of traditional unanimous requirements. To make the majority principle operational, Llull initially proposed a system of exhaustive binary comparisons that is more efficacious in producing a winner than the Condorcet system, in anticipation to the so called Copeland procedure. In contrast to some previous tentative suggestions, careful reading of Llull’s papers demonstrates that he did not propose a rank-order count system, such as those proposed later on by Cusanus and Borda. A new hypothesis is presented to explain Llull’s later proposal of an eliminatory system of partial binary comparisons. Some performance of Llull’s voting systems is estimated by innovative analysis of results in certain modern sports tournaments.     

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.     

How Social Movements Move: From First to Second Wave Developments in the UK Field of Psychiatric Contention1

In this paper I seek to unpack the notion of ‘movement’, addressing the question of what it means to say that social movements ‘move’. The concept of ‘movement’ is often used in social science to refer to change, I note, and this is clearly an appropriate usage in relationship to social movements, which often seek to bring about and/or manifest within themselves social changes. At the same time, however, movements move in the respect that the cultural forms and resources they generate are diffused (they move) across both time and space. The cultural components of a movement move in the way that, for example, a virus moves, between individuals in a ‘vulnerable’ population. The paper explores these ideas by way of an examination of the second wave of radical mental health activism in the UK.     

‘I Hate It,But It Still Sounds Good’: Collective Identity and the Evaluation of Oppositional Frame Resonance

While activists often respond to claims advanced by their opposition, little is known about how oppositional rhetoric is evaluated. This study focuses on the evaluation process, examining how movement actors assess the resonant appeal of oppositional frames. I analyze how activists in the American pro-choice movement respond to a faction of the pro-life movement that primarily frames abortion as harmful to women. Drawing on focus group conversations with pro-choice activists, I find feminist collective identity and their own experience advancing gendered frames influence which oppositional frames pro-choice actors consider most likely to resonate with a non-activist audience. These judgments subsequently guide decisions about how to respond to oppositional frames and construct of counterframes. I find activists to use collective identity to rule out potential strategies and tactics they feel are in conflict with what the group represents. I argue that in cases where similarities exist between frames and counterframes, experience advancing rhetoric superficially similar to that of the opposing movement provides strategic insight. Movement actors draw on lessons learned from their own collective framing experiences to evaluate how audiences will respond to oppositional frames with comparable cultural themes. These experiences serve as a guide, informing activists' perceptions of the frames a non-activist audience will be most likely to embrace, which frames must be addressed, and which can be safely ignored. This study emphasizes movement actors' agency and strategic decision-making processes, demonstrates how collective identity influences the framing process, and contributes to knowledge of how group experiences and identity affect perception and strategy.     

Managing the drama: the headteacher's work in an urban primary school

Previous studies of the work of primary school headteachers tend to adopt a fairly superficial approach to categorizing tasks and contacts. As part of an ethnographic study of teachers’ work in an inner city primary school, I ‘shadowed’ the headteacher for three one-week blocks plus four interviews and numerous informal talks. Life at this school was intense and often dramatic, I develop an approach which suggests the headteacher's work contains ‘routines’,‘events’ and ‘stories’ and give examples of each. Some ‘stories’ develop into ‘sagas’ or ‘social dramas’ and two instances are described, I examine how this headteacher not only copes with, but enjoys, what appears to be an increasingly stressful job.     

A Buddhist Middle Way Approach in Therapy

For many years I have been practising Buddhist meditation in my personal life. In this article, I present one way in which Buddhism has influenced my work. People who seek counselling often swing between two unsustainable or unsatisfactory alternatives, for example being ‘full on’ or doing nothing. Polarisations that trap people can be lived, remembered or imagined. Buddhism proposes the middle way as a path out of these extremes. I present ways to deconstruct dichotomies, allowing people to find more effective alternatives. I show how to actualise a ‘middle way’ approach that is appropriate to Western therapy. I will also examine times when a middle way is not an ethically acceptable response.