Mathematical structure of voting paradoxes. II: Positional voting
This paper is a companion to the paper immediately preceding it in the journal [Mathematical structure of voting paradoxes. I: Pairwise votes. Econ. Theory 15, No. 1, 1--53 (2000; Zbl 1080.91022)]. In the first paper, the author divided voting methods into positional, where the aggregation rule utilizes how each voter positions, or ranks, the various alternatives, and pairwise methods, where the rule considers majority votes among pairs of alternatives. In the earlier paper, the author primarily studied pairwise systems in comparison to the base case positional system of Borda count. Among many interesting results, he showed that the differences between the methods, and the paradoxes that arise in pairwise methods are due to certain `Condorcet effects'. Positional systems are not subject to these Condorcet effects, but can still differ in their rankings from a given profile. Here, the author analyzes these discrepancies among positional systems. The author utilizes the same profile coordinate system as in the previous paper. Central to his analysis is a concern with rankings of subsets of alternatives, that is, with Independence of Irrelevant Alternatives effects. This focus allows him to show the tremendous variety of plurality rankings that can occur, in fact, all possible combinations of rankings have supporting profiles. The core of the paper involves constructing a theory that explains how such behavior takes place. The author defines a suitable subspace that generates these effects. He shows that Borda count is the unique method that avoids profile deviation effects and that the problems besetting other positional systems are essentially due to a failure to recognize certain symmetries. The paper contains a wealth of interesting new results. The results are clearly explained and their significance brought out. There are numerous examples to illustrate the key points.
scientific article; zbMATH DE number 1415616