Arrow's single peaked domains, richness, and domains for plurality and the Borda count

TL;DR

The paper advances the study of Arrow's single-peaked domains by computationally enumerating non-isomorphic maximal domains up to $n=9$ alternatives and introducing a quantitative richness measure, showing Black's domain attains maximal richness but is not unique. It then connects these domains to independence-of-irrelevant-alternatives (IIA) analyses for different voting rules: plurality and runoff domains coincide with Arrow's single-dipped (duals of single-peaked) domains, while Borda domains under Nash's and Arrow's IIA are markedly smaller, with explicit maximum sizes characterized (e.g., $2^{n/2}$ or $2^{(n-1)/2}$ for even/odd $n$ under Nash IIA, and hierarchically cyclic structures under Arrow's IIA). The results highlight a tension between large, richly structured domains and the IIA properties of common voting rules, and provide computational tools and structural insights for exploring restricted preference domains in social choice.

Abstract

In this paper we extend the study of Arrow's generalisation of Black's single-peaked domain and connect this to domains where voting rules satisfy different versions of independence of irrelevant alternatives. First we report on a computational generation of all non-isomorphic Arrow's single-peaked domains on $n\leq 9$ alternatives. Next, we introduce a quantitative measure of richness for domains, as the largest number $r$ such that every alternative is given every rank between 1 and $r$ by the orders in the domain. We investigate the richness of Arrow's single-peaked domains and prove that Black's single-peaked domain has the highest possible richness, but it is not the only domain which attains the maximum. After this we connect Arrow's single-peaked domains to the discussion by Dasgupta, Maskin and others of domains on which plurality and the Borda count satisfy different versions of Independence of Irrelevant alternatives (IIA). For Nash's version of IIA and plurality, it turns out the domains are exactly the duals of Arrow's single-peaked domains. As a consequence there can be at most two alternatives which are ranked first in any such domain. For the Borda count both Arrow's and Nash's versions of IIA lead to a maximum domain size which is exponentially smaller than $2^{n-1}$, the size of Black's single-peaked domain.

Theorems & Definitions (33)

• Definition 1
• Lemma 1
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• Proposition 1
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• Lemma 2
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• Corollary 1
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