Matrix Rationalization via Partial Orders

TL;DR

This paper explores how preference matrices can be rationalized by voters with partial orders of bounded width, generalizing the classical total order case, and relates this to graph coloring parameters.

Contribution

It introduces the concept of rationality number for preference matrices and characterizes it using graph parameters like chromatic and dichromatic numbers.

Findings

For half-integral matrices, the rationality number relates to the chromatic number of the unanimity graph.

For integral matrices, the rationality number relates to the dichromatic number of the voting graph.

Provides a new framework connecting preference rationalization with graph coloring concepts.

Abstract

A preference matrix $M$ has an entry for each pair of candidates in an election whose value $p_{ij}$ represents the proportion of voters that prefer candidate $i$ over candidate $j$. The matrix is rationalizable if it is consistent with a set of voters whose preferences are total orders. A celebrated open problem asks for a concise characterization of rationalizable preference matrices. In this paper, we generalize this matrix rationalizability question and study when a preference matrix is consistent with a set of voters whose preferences are partial orders of width $\alpha$. The width (the maximum cardinality of an antichain) of the partial order is a natural measure of the rationality of a voter; indeed, a partial order of width $1$ is a total order. Our primary focus concerns the rationality number, the minimum width required to rationalize a preference matrix. We present two main results. The first concerns the class of half-integral preference matrices, where we show the key parameter required in evaluating the rationality number is the chromatic number of the undirected unanimity graph associated with the preference matrix $M$. The second concerns the class of integral preference matrices, where we show the key parameter now is the dichromatic number of the directed voting graph associated with $M$.