Sven Berg

Condorcet's jury theorem, dependency among jurors

With the aid of a simple discrete probability model, the Condorect Jury Theorem is extended to situations where there is dependency between judgement. It is shown that negative intra-voter correlation improves on jury competence in the sense that the probability of a correct decision increases. Positive correlation, on the other hand, tends to decrease jury competence. The results given here are obtained with the aid of simple recursion relations.

Paradox of voting under an urn model: The effect of homogeneity

We propose a simple Pólya-variety urn model for calculating paradox-of-voting probabilities. The model contains a homogeneity parameter, and for specific values of this parameter the model reduces to cases previously discussed in the literature. We derive a Dirichlet family of distributions for describing the assignment of preference profiles in large committees, and we show how the homogeneity parameter relates to measures of similarity among voters, suggested in prior studies.

Continuity-of-care Measures: An Analytic and Empirical Comparison

The authors compared four different continuity-of-care measures recently suggested in the literature. First, an analytic comparison is made and useful inequalities are derived. Then the continuity indexes are applied to an extensive data material from a Swedish ambulatory care center. The four indexes give fairly similar results, and the basic finding is the substantially higher index values for elderly patients. Finally, computed index values are compared with what could be expected under a scheme of random assignment of patients to doctors

Randomized decision rules in voting games: a model for strict proportional power

The concept of strict proportional power is introduced, as a means of formalizing a desire to avoid discrepancy between the seat distribution in a voting body and the actual voting power in that body, as measured by power indices in common use. Proportionality is obtained through use of a randomized decision rule (majority rule). Some technical problems which arise are discussed in terms of simplex geometry. Practical implications and problems in connection with randomized decision rules are indicated.

A note on the paradox of voting: Anonymous preference profiles and May's formula

Since Black (1958: 50-51), a discrete literature has developed devoted to the calculation of probabilities for the occurrence of the voting paradox (Williamson and Sargent, 1967; Niemi and Weisberg, 1968; Garman and Kamien, 1968; Inada, 1969; de Meyer and Plott, 1970; Bjurulf, 1972; Jamison and Luce, 1972; Fine, 1973; Kelly, 1974; and more recently, Brams, 1976; Gehrlein and Fishburn, 1976; and Fishburn and Gehrlein, 1980). Kelly (1974) finds three main strands in this literature. First, some researchers report exact probabilities or percentages for small numbers of voters and alternatives. Second, in more intractable cases, machine simulation creates estimates of paradox probabilities (Bjurulf, 1972). Finally, there have been attempts to find explicit formulas, which, as a rule, have been wildly intractable. Common to all of these approaches is a discussion of the voting paradox using one of two basic probability models. The first model assumes that voters are identifiable. The second assumes that they are anonymous. The first model is the one most often found in the literature. However, Kuga and Nagatani (1974) and, in particular, Gehrlein and Fishburn (1976) derive results based on the second model.

COLLECTIVE DECISION MAKING IN HIERARCHIES

Abstract This paper investigates properties of composite majority functions with the aid of recursive formulas in the context of collective decision making in multi-tier hierarchies. The effects of changes in system structure on collective competence are examined. In addition to the theoretical importance, the results given here are applicable in designing hierarchies where the cost in the form of competence loss has to be considered against the benefit of time-saving and other expenses.

Condorcet's Jury Theorem and the reliability of majority voting

The effect on the Jury Theorem of dependency among votes is discussed. Condorcets original model and theorem depend crucially on the assumption of independence and the applicability of the binomial distribution. Two simple extensions of the binomial distribution are used to illustrate the effects of dependency on the quality of group decision making. With the correlated binomial model, it is possible to isolate the effect of pairwise dependency. In the presence of fairly strong pairwise dependency, we are not even guaranteed the natural property of monotonicity with respect to voters. A Pólya-Eggenberger model illustrates the effect of contagion on group competence. A special case of the beta-binomial distribution is used to demonstrate that, even in the presence of synergetic group effects, we are not guaranteed infallible decisions from a very large group. Consequences for an epistemic theory of democracy are indicated.

A note on plurality distortion in large committees

Abstract Simple plurality voting allows the somewhat surprising possibility that a majority of a committee or an electorate may prefer one of the defeated alternatives to the plurality winner. This paradox is distinct from Condorcets paradox and has received less attention in the literature. Recently, however, the frequency of the Borda effect anticipated under different cultures has been computed and some empirical results have also been reported. Here results are obtained for both impartial and partial cultures using a class of Polya urn models. Emphasis is on asymptotic investigations and results valid for large committees.

Evaluation of some weighted majority decision rules under dependent voting

Abstract This paper studies dichotomous voting with correlated votes and the effect of the dependency on a special class of weighted majority rules, containing, for example, two expert rules and a simple majority rule. To do this, I introduce a model for conditional skill probabilities dependent upon the opinion and skill of an influential group member. A certain level of correlation between votes is shown to lead to minimum collective competence under the decision rules studied, while negative dependency may lead to increased collective decisional efficiency.

Indirect voting systems: Banzhaf numbers, majority functions and collective competence

Abstract This paper discusses the reliability of group judgment in the context of indirect voting systems and the Condorcet jury theorem. To describe such systems, composite majority functions are introduced and their properties studied. Associated Banzhaf numbers appear as a natural measure of decisional power. The paper examines how collective judgmental competence depends on structural properties of the decision making system. Formulas are developed to measure the loss (or gain) in judgmental competence due to simple changes in system structure, such as overlapping membership.