Borda Rule Research Articles

The Condorcet Efficiency of Voting Rules with Mutually Coherent Voter Preferences: A Borda Compromise

The Condorcet Efficiency of a voting rule is defined as the conditional probability that the voting rule elects the Pairwise Majority Rule Winner (PMRW), given that a PMRW exists. Five simple voting rules are considered in this paper: Plurality Rule, Negative Plurality Rule, Borda Rule, Plurality Elimination Rule and Negative Plurality Elimination Rule. In order to study the impact that the presence of degrees of group mutual coherence in voting situations will have on the probability of selecting the PMRW for each of these rules, we develop representations for their Condorcet Efficiency as a function of the proximity of voters' preferences on candidates to being perfectly singlepeaked, perfectly single-troughed or perfectly polarized. The results we obtain lead us to appeal for a Borda Compromise.

A Social Choice Analysis of the Borda Rule in a General Linguistic Framework

AbstractIn this paper the Borda rule is extended by allowing the voters to show their preferences among alternatives through linguistic labels. To this aim, we need to add them up for assigning a qualification to each alternative and then to compare such qualifications. Theoretically, all these assessments and comparisons fall into a totally ordered commutative monoid generated by the initial set of linguistic labels. Practically, we show an example which illustrates the suitability of this linguistic approach. Finally, some interesting properties for this Borda rule are proven in the Social Choice context.

A Social Choice Analysis of the Borda Rule in a General Linguistic Framework

In this paper the Borda rule is extended by allowing the voters to show their preferences among alternatives through linguistic labels. To this aim, we need to add them up for assigning a qualification to each alternative and then to compare such qualifications. Theoretically, all these assessments and comparisons fall into a totally ordered commutative monoid generated by the initial set of linguistic labels. Practically, we show an example which illustrates the suitability of this linguistic approach. Finally, some interesting properties for this Borda rule are proven in the Social Choice context.

Condorcet vs. Borda in light of a dual majoritarian approach

Many voting rules and, in particular, the plurality rule and Condorcet-consistent voting rules satisfy the simple-majority decisiveness property. The problem implied by such decisiveness, namely, the universal disregard of the preferences of the minority, can be ameliorated by applying unbiased scoring rules such as the classical Borda rule, but such amelioration has a price; it implies erosion in the implementation of the widely accepted “majority principle”. Furthermore, the problems of majority decisiveness and of the erosion in the majority principle are not necessarily severe when one takes into account the likelihood of their occurrence. This paper focuses on the evaluation of the severity of the two problems, comparing simple-majoritarian voting rules that allow the decisiveness of the smallest majority larger than 1/2 and the classical Borda method of counts. Our analysis culminates in the derivation of the conditions that determine, in terms of the number of alternatives k, the number of voters n, and the relative (subjective) weight assigned to the severity of the two problems, which of these rules is superior in light of the dual majoritarian approach.

A RULE FOR AGGREGATING INDIVIDUAL WELFARE MEASURES

It is common, but often incorrect, practice in most applied works, especially in the field of applied general equilibrium models, to evaluate the overall impact of alternative policy changes by means of the sum of equivalent variations of (different classes of) individual consumers. This note proposes to use these same equivalent variations in a different way, namely, to aggregate them by the Borda rule.

Defining the Borda count in a linguistic decision making context

Different kinds of decision rules have been successfully implemented under a linguistic approach. This paper aims the same goal for the Borda count, a well-known procedure with some interesting features. In order to this, two ways of extension from the Borda rule to a linguistic framework are proposed taking into account all the agents’ opinions or only the favorable ones for each alternative when compared with each other. In the two cases, both individual and collective Borda counts are analyzed, asking for properties as good as those of the original patterns.

Does Imitation Benefit Cue Order Learning?

Inferences are often based on uncertain cues, and the accuracy of such inferences depends on the order in which the cues are searched. Previous research has shown that people and computers progress only slowly in individual learning of cue orderings through feedback. A clue to how people (as opposed to computers) solve this problem is social learning: By exchanging information with others, people can learn which cues are relevant and the order in which they should be considered. By means of simulation, we demonstrate that imitate-the-best and imitate-the-majority speed up individual learning, whereas a third social rule, the Borda rule, does not. Imitate-the-best also leads to a steep increase in learning after a single social exchange, to cue orders that are more accurate than ecological validity, and to faster learning than when individuals gain the learning experience of all other group members but learn without social exchange. In two experiments, we find that people speed up cue learning in a similar way when provided with social information, both when they obtain the information from the experimenter or in free discussions with others.

Borda and the maximum likelihood approach to vote aggregation

It is shown that, for a suitably chosen probability function, the maximum likelihood rule for vote aggregation coincides with the Borda rule, thus, partially reconciling the Borda and the Condorcet methods.

A characterization of the Borda rule in peer ratings

We consider the model of peer ratings, where agents are requested to report preferences on the other agents and a social welfare function aggregates such preferences into a social preference. We focus on the class of social welfare functions called the generalized scoring rules and characterize the Borda rule as the unique generalized scoring rule satisfying the “Pareto principle”.

The Borda rule is also intended for dishonest men

This paper examines the welfare consequences of strategic voting under the Borda rule in a comparison of utilitarian efficiencies in simulated voting games under two behavioural assumptions: expected utility-maximising behaviour and sincere behaviour. Utilitarian efficiency is higher in the former than in the latter. Strategic voting increases utilitarian efficiency particularly if the distribution of preference intensities correlates with voter types. The Borda rule is shown to have two advantages: strategic voting is beneficial even if some but not all voter types engage in strategic behaviour, and even if the voters’ information is based on unreliable signals.

A comparison between the prudent order and the ranking obtained with Borda's, Copeland's, Slater's and Kemeny's rules

Arrow and Raynaud suggested that the result of a ranking rule should be a prudent order. We prove that we can construct profiles of linear orders for which the unique prudent order is the exact opposite of the ranking obtained with Borda's rule or Copeland's rules. Furthermore, we show that we can construct profiles of linear orders such that the unique prudent order winner can be found at any position in the corresponding unique order found by Slater's or Kemeny's rules. Finally, we show that there exist profiles where the unique Slater or Kemeny order is the exact opposite of one prudent order.

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 (sales- man) 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 investi- gate 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.

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 $$\lfloor m/2\rfloor$$ -approval voting when m ≥ 5.

The Costs of Implementing the Majority Principle: The Golden Voting Rule

In a context of constitutional choice of a voting rule, this paper presents an economic analysis of scoring rules that identifies the golden voting rule under the impartial culture assumption. This golden rule depends on the weights β and (1−β) assigned to two types of costs: the cost of majority decisiveness (‘tyranny’) and the cost of the ‘erosion’ in the majority principle. Our first main result establishes that in voting contexts where the number of voters n is typically considerably larger than the number of candidates k, the golden voting rule is the inverse plurality rule for almost any positive β. Irrespective of n and k, the golden voting rule is the inverse plurality rule if β ≥ 1/2 .. This hitherto almost unnoticed rule outperforms any other scoring rule in eliminating majority decisiveness. The golden voting rule is, however, the plurality rule, the most widely used voting rule that does not allow even the slightest ‘erosion’ in the majority principle, when β=0. Our second main result establishes that for sufficiently “small size” voting bodies, the set of potential golden rules consists at most of just three rules: the plurality rule, the Borda rule and the inverse plurality rule. On the one hand, this finding provides a new rationalization to the central role the former two rules play in practice and in the voting theory literature. On the other hand, it provides further support to the inverse plurality rule; not only that it is the golden rule in voting contexts, it also belongs, together with the plurality rule and the Borda method of counts, to the “exclusive” set of potential golden voting rules in small committees.

Scoring rules: an alternative parameterization

This note presents an alternative parameterization of any scoring rule that satisfies the score-expansion property. This parameterization is based on the vector that specifies, for every number of alternatives k, k ≥ 3, the minimal size of a coalition that can veto an alternative which is preferred by everybody outside the coalition. Our result sheds new light on the commonly used plurality and Borda rules, as well as the inverse plurality rule and any “vote for t alternatives rule”.

Minimal monotonic extensions of scoring rules

Noting the existence of social choice problems over which no scoring rule is Maskin monotonic, we characterize minimal monotonic extensions of scoring rules. We show that the minimal monotonic extension of any scoring rule has a lower and upper bound, which can be expressed in terms of alternatives with scores exceeding a certain critical score. In fact, the minimal monotonic extension of a scoring rule coincides with its lower bound if and only if the scoring rule satisfies a certain weak monotonicity condition (such as the Borda and antiplurality rule). On the other hand, the minimal monotonic extension of a scoring rule approaches its upper bound as its degree of violating weak monotonicity increases, an extreme case of which is the plurality rule with a minimal monotonic extension reaching its upper bound.

Borda and Condorcet: Some Distance Results

This paper provides a distance based analysis of the Borda rule with respect to Condorcet’s criterion. It shows that the minimal Condorcet consistency present in the Borda rule, whenever a Condorcet winner (the alternative that wins against every other alternative in a pairwise contest) exists, disappears in the case of voting cycles. First, it is shown that for certain preference profiles the Borda winner is furthest from being a Condorcet winner. Second, it is shown that there exist preference profiles for which the Borda winner is closest from being a Condorcet loser (the alternative that loses against every other alternative in a pairwise contest).

On the Closeness Aspect of Three Voting Rules: Borda – Copeland – Maximin

This paper provides a comparison of three voting rules, the Borda rule, the Copeland rule and the maximin rule. Our analysis based on the rankings derived from those voting rules will shed new light on existing comparisons based on “closeness” between the three voting rules under investigation. In particular it will be shown that both, the rankings derived from the Borda rule and the maximin rule and the rankings derived from the Copeland rule and the maximin rule, will be exactly the opposite for certain preference profiles.

The Dodgson ranking and the Borda count: a binary comparison

This paper provides a binary comparison of two preference aggregation rules, the Borda rule and Dodgson's rule. Both of these rules guarantee a transitive ranking of the alternatives for every list of individual preferences and therefore avoid the problem of voting cycles. It will be shown that for certain lists of individual preferences the rankings derived from the Borda rule and Dodgson's rule are antagonistic.

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 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 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.