An analysis of simple counting methods for ordering incomplete ordinal data

Measurement in the social sciences often involves an attempt to completely order a set of entities on the basis of an underlying attribute. However, limitations of the measurement process often prevent complete empirical determination of the desired ordering. Nevertheless, the ordinal data obtained from the measurement process can be used in attempting to recover or construct more of the underlying order than is provided by the data. Previous research (Fishburn and Gehrlein, 1974a) has shown that a simple one-stage construction method, referred to as the cardinal rule, is fairly effective in correctly identifying ordered pairs in the underlying linear order that are not identified by the measurement process. The present paper re-examines the cardinal rule from the perspective of construction methods based on simple counting measures derived from the data, and argues that it is the best one-stage method in this class when a natural monotonicity assumption holds for the measurement process. The paper then examines two-stage construction rules that are based on the cardinal rule and the simple counting measures. It is shown that one of the two-stage rules gives better overall results than does the cardinal rule by itself.     

A representation for quadrivariate normal positive orthant probabilities

For four variables x₁,x₂, x₃ and x₄, which have a quadrivariate normal distribution with means equal to zero, the positive ortrhant probability is the probability that all of the x.'s are simultaneously positive. A representation for the quadrivariate normal positive orthant probability is obtained and it is a function of no more than three integrals over a single variable. Extensive testing has shown this representation to be very efficient on a computational basis.     

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.     

Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*

Many studies have considered the probability that a pairwise majority rule (PMR) winner exists for three candidate elections. The absence of a PMR winner indicates an occurrence of Condorcet's Paradox for three candidate elections. This paper summarizes work that has been done in this area with the assumptions of: Impartial Culture, Impartial Anonymous Culture, Maximal Culture, Dual Culture and Uniform Culture. Results are included for the likelihood that there is a strong winner by PMR, a weak winner by PMR, and the probability that a specific candidate is among the winners by PMR. Closed form representations are developed for some of these probabilities for Impartial Anonymous Culture and for Maximal Culture. Consistent results are obtained for all cultures. In particular, very different behaviors are observed for odd and even numbers of voters. The limiting probabilities as the number of voters increases are reached very quickly for odd numbers of voters, and quite slowly for even numbers of voters. The greatest likelihood of observing Condorcet's Paradox typically occurs for small numbers of voters. Results suggest that while examples of Condorcet's Paradox are observed, one should not expect to observe them with great frequency in three candidate elections.     

The sensitivity of weight selection for scoring rules to profile proximity to single-peaked preferences

Niemi (Am Polit Sci Rev 63:488–497, 1969) proposed a simple measure of the cohesiveness of a group of n voters’ preferences that reflects the proximity of their preferences to single-peakedness. For three-candidate elections, this measure, k, reduces to the minimum number of voters who rank one of the candidates as being least preferred. The current study develops closed form representations for the conditional probability, PASW(n,IAC|k), that all weighted scoring rules will elect the Condorcet winner in an election, given a specified value of k. Results show a very strong relationship between PASW(n,IAC|k) and k, such that the determination of the voting rule to be used in an election becomes significantly less critical relative to the likelihood of electing the Condorcet winner as voters in a society have more structured preferences. As voters’ preferences become more unstructured as measured by their distance from single-peakedness, it becomes much more likely that different voting rules will select different winners.A preliminary version of this paper was presented at the European Public Choice Society Conference in Berlin, Germany, April 15–18, 2004.     

Division of power in legislatures with two cohesive subgroups

We formulate a voting model for a legislature composed of two disjoint and cohesive subgroups. Each subgroup has a leader, and its cohesiveness is measured by the probability that a member votes the same way as the leader, which is presumed to be the same for all non-leader members within a subgroup. We examine the relative powers of the subgroups as a function of their sizes and cohesiveness.     

The power of a cohesive subgroup within a voting body

We examine the power of a subgroup within a voting body as a function of its size and cohesiveness. The measure of cohesiveness is the probability that an arbitrary member of the subgroup will vote the same way as the leader of the subgroup. All voters are assumed to vote independently, and those not in the subgroup vote for or against with probability 1/2 in each case. The power of the subgroup is the probability that its vote differential exceeds the vote differential of the members of the body not in the subgroup. Tradeoffs between relative subgroup size and its follow-the-leader probability are considered.     

Condorcet efficiency: A preference for indifference

The Condorcet winner in an election is the candidate who would be able to defeat all other candidates in a series of pairwise elections. The Condorcet efficiency of a voting procedure is the conditional probability that it will elect the Condorcet winner, given that a Condorcet winner exists. The study considers the Condorcet efficiency of weighted scoring rules (WSR's) on three candidates for large electorates when voter indifference between candidates is allowed. It is shown that increasing the proportion of voters who have partial indifference will increase the probability that a Condorcet winner exists, and will also increase the Condorcet efficiency of all WSR's. The same observation is observed when the proportion of voters with complete preferences on candidates is reduced. Borda Rule is shown to be the WSR with maximum Condorcet efficiency over a broad range of assumptions related to voter preferences. The result of forcing voters to completely rank all candidates, by randomly breaking ties on candidates that are viewed as indifferent, leads to a reduction in the probability that a Condorcet winner exists and to a reduction in the Condorcet efficiency of all WSR's. Received: 31 July 1999/Accepted: 11 February 2000     

Peter C. Fishburn (1936–2021)

Theory and Decision -