Armajac Raventós Pujol

March 3, 2023

Title: Simplicial complexes and the index lemma: A pathway to reach agreements fairly

Abstract: Aggregating individual preferences is a fundamental problem in democracy: How can we take collective decisions fairly based on individual preferences? Arrow's impossibility theorem (1951) proves that it is not possible to do it when we assume some apparently mild conditions. Fortunately, in some cases, aggregation is possible when the domain of individual preferences is restricted. That is, when voters can only report some preferences, good aggregation rules exist. However, no theorem characterizes the domains in which aggregation is possible, and the problem remains open.

Despite the Arrovian model being purely combinatorial, Baryshnikov (1993) used simplicial complexes and homology to prove Arrow's theorem and exposed a conjecture which characterized restricted domains through homology groups. The main drawback of using homology is that it is not affordable for most of the social scientists. Therefore, instead of homology, we have used combinatorial topology tools such as the Index Lemma (the combinatorial counterpart to Poincare's Lemma) to tackle the problem. First, we have proved the Arrow's impossibility theorem, showing that combinatorial topology is helpful for our purposes.

Second, we have characterized the domains allowing aggregation rules for the base case of two voters and three candidates. Our characterization proves that homology groups are not enough to characterize such domains. Our result gives us hope to obtain a general characterization of the good domains for aggregating preferences. Moreover, it could be implemented computationally, making it handled by practitioners in politics and economics.

Keywords: social choice, combinatorial topology, Arrow's impossibility theorem, index lemma, domain restriction, aggregating preferences.