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.     

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.     

The sensitivity of weight selection on the Condorcet efficiency of weighted scoring rules

A weighted scoring rule, Rule λ, on three alternative elections selects the winner by awarding 1 point to each voter's first ranked candidate, λ points to the second ranked candidate, and zero to the third ranked candidate. The Condorcet winner is the candidate that would defeat each other candidate in a series of pairwise elections by majority rule. The Condorcet efficiency of Rule λ is the conditional probability that Rule λ selects the Condorcet winner, given that a Condorcet winner exists. Borda rule (λ=1/2) is the weighted scoring rule that maximizes Condorcet efficiency. The current study considers the conditional probability that Borda rule selects the Rule λ winner, given that Rule λ elects the Condorcet winner with a large electorate. Received: 21 August 1996 / Accepted: 7 January 1997     

Does majoritarian approval matter in selecting a social choice rule? An exploratory panel study

This study is an attempt to empirically detect the public opinion concerning majoritarian approval axiom. A social choice rule respects majoritarian approval iff it chooses only those alternatives which are regarded by a majority of “voters” to be among the “better half” of the candidates available. We focus on three social choice rules, the Majoritarian Compromise, Borda’s Rule and Condorcet’s Method, among which the Majoritarian Compromise is the only social choice rule always respecting majoritarian approval. We confronted each of our 288 subjects with four hypothetical preference profiles of a hypothetical electorate over some abstract set of four alternatives. At each hypothetical preference profile, two representing the preferences of five and two other of seven voters, the subject was asked to indicate, from an impartial viewpoint, which of the four alternatives should be chosen whose preference profile was presented, which if that is unavailable, then which if both of the above are unavailable, and finally which alternative should be avoided especially. In each of these profiles there is a Majoritarian Compromise-winner, a Borda-winner and a Condorcet-winner, and the Majoritarian Compromise-winner is always distinct from both the Borda-winner and the Condorcet-winner, while the Borda- and Condorcet-winners sometimes coincide. If the Borda- and Condorcet-winners coincide then there are two dummy candidates, otherwise only one, and dummies coincide with neither of the Majoritarian Compromise-, Borda- or Condorcet-winner. We presented our subjects with various types of hypothetical preference profiles, some where Borda respecting majoritarian approval, some where it failed to do so, then again for Condorcet, some profiles it respected majoritarian approval and some where it did not. The main thing we wanted to see was whether subjects’ support for Borda and Condorcet was higher when this social choice rule respected majoritarian approval than it did not. Our unambiguous overall empirical finding is that our subjects’ support for Borda and Condorcet was significantly stronger as they respect majoritarian approval.     

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     

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.     

Manipulability measures of common social choice functions

All social choice functions are manipulable when more than two alternatives are available. I evaluate the manipulability of the Borda count, plurality rule, minimax set, and uncovered set. Four measures of manipulability are defined and computed stochastically for small numbers of agents and alternatives.  Social choice rules derived from the minimax and uncovered sets are found to be relatively immune to manipulation whether a sole manipulating agent has complete knowledge or absolutely no knowledge of the preferences of the others. The Borda rule is especially manipulable if the manipulating agent has complete knowledge of the others. Received: 5 January 1996/Accepted: 31 July 1998     

Scoring rules on dichotomous preferences

In this paper, we study individual incentives to report preferences truthfully for the special case when individuals have dichotomous preferences on the set of alternatives and preferences are aggregated in form of scoring rules. In particular, we show that (a) the Borda Count coincides with Approval Voting, (b) the Borda Count is the only strategy-proof scoring rule, and (c) if the size of the electorate is greater than three, then the dichotomous preference domain is the unique maximal rich domain under which the Borda Count is strategy-proof. I thank Jordi Massó for his supervision and his never-ending encouragement. Miguel-ángel Ballester showed me how to improve on earlier drafts of the paper. Salvador Barberà, Carmen Bevía, Bhaskar Dutta, Lars Ehlers, Alejandro Neme, Shmuel Nitzan and Yves Sprumont helped me a lot with their comments. All remaining errors are mine. This research was undertaken with support from the fellowship 2001FI 00451 of the Generalitat de Catalunya and from the research grant BEC2002-02130 of the Ministerio de Ciencia y Tecnología of Spain while I have been a graduate student at Universitat Autònoma de Barcelona.     

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.     

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.     

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.     

MANIPULATING VOTING PROCEDURES

A voting procedure can be manipulated if, by misrepresenting his preferences, some individual can secure an outcome which he prefers to the outcome he gets when he is honest.
This is an expository paper on the theory of voting manipulation. Section I is an historical sketch of the contributions of Condorcet, de Borda, Arrow, and others. Section II provides a set of examples of manipulation: of plurality voting, of majority voting, of exhaustive voting, of the single transferable vote procedure, and of approval voting. It also contains an example of a nonmanipulable random voting scheme. Section HI provides a simple proof of the Gibbard-Satterthwaite manipulation theorem.     

Social choice by majority rule with rational participation

This paper explores the social choice properties of majority rule when voters rationally participate in elections. Employing the basic model of voter behavior developed originally by Ledyard (1984) and Palfrey and Rosenthal (1985), we show that when the electorate is sufficiently large, a participation equilibrium exists and is unique. The main result of the paper shows that under these conditions the social preference ordering induced by majority rule is identical to that given by the expected utility of a randomly selected voter, implying the existence of a Condorcet point in the proposal space. A final section provides intuition for the main theorems and relates the equilibrium of this majority rule game to the median voter result.     

A Borda count for partially ordered ballots

The application of the theory of partially ordered sets to voting systems is an important development in the mathematical theory of elections. Many of the results in this area are on the comparative properties between traditional elections with linearly ordered ballots and those with partially ordered ballots. In this paper we present a scoring procedure, called the partial Borda count, that extends the classic Borda count to allow for arbitrary partially ordered preference rankings. We characterize the partial Borda count in the context of weighting procedures and in the context of social choice functions.     

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     

Are Condorcet procedures so bad according to the reinforcement axiom?

A Condorcet social choice procedure elects the candidate that beats every other candidate under simple majority when such a candidate exists. The reinforcement axiom roughly states that given two groups of individuals, if these two groups select the same alternative, then this alternative must also be selected by their union. Condorcet social choice procedures are known to violate this axiom. Our goal in this paper is to put this important voting theory result into perspective. We then proceed by evaluating how frequently this phenomenon is susceptible to occur.     

Unequivocal majority and Maskin-monotonicity

The unequivocal majority of a social choice rule is a number of agents such that whenever at least this many agents agree on the top alternative, then this alternative (and only this) is chosen. The smaller the unequivocal majority is, the closer it is to the standard (and accepted) majority concept. The question is how small can the unequivocal majority be and still permit the Nash-implementability of the social choice rule; i.e., its Maskin-monotonicity. We show that the smallest unequivocal majority compatible with Maskin-monotonicity is n- ë \fracn-1m û{n-\left\lfloor \frac{n-1}{m} \right\rfloor} , where n ≥ 3 is the number of agents and m ≥ 3 is the number of alternatives. This value is equal to the minimal number required for a majority to ensure the non-existence of cycles in pairwise comparisons. Our result has a twofold implication: (1) there is no Condorcet consistent social choice rule satisfying Maskin-monotonicity and (2) a social choice rule satisfies k-Condorcet consistency and Maskin-monotonicity if and only if k 3 n- ë \fracn-1m û{k\geq n-\left\lfloor \frac{n-1}{m}\right\rfloor}.     

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.     

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     

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.