Preference aggregation theory without acyclicity: The core without majority dissatisfaction

TL;DR

This paper explores preference aggregation without the usual acyclicity assumption, focusing on maximal elements within fixed agendas, and characterizes conditions under which the core without majority dissatisfaction remains nonempty.

Contribution

It introduces a new approach replacing acyclicity with maximal elements assumption and characterizes the nonemptiness of the core without majority dissatisfaction in terms of the Nakamura number.

Findings

The core is nonempty iff the number of alternatives is less than the Nakamura number.

The core without majority dissatisfaction depends only on players' maximal elements.

The core without majority dissatisfaction behaves better in extended frameworks.

Abstract

Acyclicity of individual preferences is a minimal assumption in social choice theory. We replace that assumption by the direct assumption that preferences have maximal elements on a fixed agenda. We show that the core of a simple game is nonempty for all profiles of such preferences if and only if the number of alternatives in the agenda is less than the Nakamura number of the game. The same is true if we replace the core by the core without majority dissatisfaction, obtained by deleting from the agenda all the alternatives that are non-maximal for all players in a winning coalition. Unlike the core, the core without majority dissatisfaction depends only on the players' sets of maximal elements and is included in the union of such sets. A result for an extended framework gives another sense in which the core without majority dissatisfaction behaves better than the core.