Voting paradoxes and group coherence. The Condorcet efficiency of voting rules.
On the one hand there is the theoretical literature giving many examples of voting situations in which a particular voting mechanism yields a counterintuitive outcome, Condorcet's paradox being the most famous of these paradoxical results. On the other hand there are many empirical studies pointing out that these paradoxes hardly occur in practical voting situations. The goal of the authors throughout their book is to integrate the theoretical results obtained from formal probability representations with empirical results from other studies. Their overall conclusion is that `while Borda Rule is not always the most effective voting rule for selecting the Pairwise Majority Rule Winner in all scenarios, it is resistant to the potential problem of performing very poorly. Moreover, scenarios do exist for all other common voting rules in which the possible outcome of very poor performance is a significant issue.' In Chapter 1 a number of voting paradoxes (among which Condorcet's, Borda's, no show, Ostrogorski's), are introduced and empirical studies are summarized to indicate that some of the most common paradoxes are relatively unlikely to be observed in actual elections. The authors use several conditions, Dual Culture, Impartial Culture, Impartial Anonymous Culture and Maximal Culture to obtain representations for the probability that voting events will be observed and use these mathematical models to suggest that voting paradoxes should become less likely as voters' preferences exhibit greater degrees of homogeneity, or as voters' preferences on candidates become more statistically dependent. In Chapter 2 the authors focus on the association between the likelihood that a Pairwise Majority Rule Winner exists and degrees of social homogeneity, where they distinguish between Population Specific Measures of Homogeneity and Situation Specific Measures of Homogeneity. They conclude among others that `Condorcet's paradox should very rarely be observed in any real elections on a small number of candidates with large electorates, as long as voters' preferences reflect any reasonable degree of group mutual coherence from a number of different models, and the observations that have been made from numerous empirical studies should no longer seem surprising.' More precisely, the likelihood that Condorcet's paradox is observed consistently decreases as the degree of group mutual coherence increases. Chapter 3 tries to extend the observations for Condorcet's Paradox in Chapter 2 to Condorcet's so called Other Paradox and to Borda's Paradox. It turns out that the probability of observing Condorcet's Other Paradox is very small under the assumption of Impartial Anonymous Culture. However, the authors show that the connection between group mutual coherence and the likelihood that Borda's Paradox is observed does not have such a clear outcome: specific conditions can be defined so that a high likelihood exists for observing Borda's Paradox. However, the overall probability (with Impartial Anonymous Culture) that Borda's Paradox will be observed is rather small! The results of Chapter 4 show that it is very unlikely that any extreme form of Ostrogorski's Paradox would ever be observed in any real situation with large electorates that showed any significant degree of group mutual coherence. The probabilities for the Monotonicity Paradox and the No Show Paradox, although higher than those for Ostrogorski's Paradox, turn out to be typically not very large. In Chapter 5 the authors consider the Condorcet Efficiency of voting rules and discuss the Efficiency Hypothesis: as voters have preferences on candidates that reflect increased levels of social homogeneity or group mutual coherence, voting rules should tend to show an increased level of Condorcet Efficiency. After an extensive analysis with a number of different measures of group mutual coherence in Chapter 6, it turns out that there is very little general support for the Efficiency Hypothesis. However, the authors make a number of general observations regarding the relative performance of voting rules on the basis of Condorcet Efficiency for three candidate elections. The authors argue that the Borda Compromise is a rather easy conclusion to reach in the evaluation of the single-stage voting rules. This Borda Compromise will use the Borda Rule when nothing is known a priori about the type of model that is likely to reflect the preferences of the electorate. This compromise position will have a high likelihood of selecting the Pairwise Majority Rule Winner. In Chapter 7 the authors provide numerous other pieces of evidence to support the general notion that while Borda's Rule will not necessarily always produce the best election outcome, it can typically be expected to perform very well relative to any voting rule on a number of different criteria. They also provide evidence to indicate that reports on the commonly held belief that the Borda Rule is highly susceptible to strategic manipulation appear to be greatly exaggerated. In Chapter 8 two scenarios are found to exist that produce very high probabilities of observing voter situations for which all Weighted Scoring Rules will elect the Pairwise Majority Rule Winner. At the same time other scenarios are given that give a very low probability of such an outcome. Approval Voting is shown to have many advantages that are all heavily dependent on the assumption of dichotomous voters' preferences; however, Approval Voting shows weakness when compared to Constant Scoring Rules, and particularly to the Borda Rule, on the basis of Condorcet Efficiency. In the last Chapter 9 it is pointed out that the Condorcet Ranking Efficiency of the Borda Rule remains somewhat stable across the complete range of all measures of group mutual coherence. The Borda Rule generally dominates both Plurality Rule and the Negative Plurality Rule for all weak and strong measures of group mutual coherence. Finally, the authors pay attention to the evaluation of voting rules on their ability to select a committee, rather than simply select a single winner. This book bridges the gap between the theoretical literature showing the possibility of counterintuitive results and the empirical findings of many experiments in which these counterintuitive results hardly or not at all appear. This is a very well written book, extremely rich in information, integrating many existing and new results and with an impressive number of references to the literature. It is an important and relevant step forward in the Theory of Social Choice, highly recommended to anyone interested in this field.
scientific article; zbMATH DE number 5865107