Matrix Rationalization via Partial Orders

TL;DR

The paper investigates when a pairwise preference matrix $M=(p_{ij})$ can be explained by a population of voters whose preferences are partial orders of bounded width $\alpha$, introducing the rationality number $\alpha(M)$. It provides a sharp dichotomy: for half-integral matrices, $\alpha(M)$ is controlled by the chromatic number $\chi(G_M)$ of the unanimity graph, with $(1/5)\chi(G_M) \le_\exists \alpha(M) \le_\forall \chi(G_M)$; for integral matrices, $\alpha(M)$ equals the dichromatic number $\overrightarrow{\chi}(D_M)$ of the voting graph, with bounds $\chi(G_M)/(2\log n+1) \le_\exists \alpha(M) \le_\forall 3\chi(G_M)/\log n$ (where $n$ is the number of candidates). The authors prove a series of structural results, upper and lower bounds, and exact characterizations in the integral case, linking rationalizability to classical graph-coloring concepts and presenting NP-completeness results for computing $\alpha(M)$. These findings illuminate the complexity of explaining observed choice data and suggest graph-theoretic avenues for designing efficient approximations or heuristics. Overall, the work connects social-choice rationalizability with chromatic and dichromatic coloring, providing both theoretical characterizations and hardness results relevant to data analysis and algorithm design.

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 $α$. 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 $M$.

Figures (8)

• Figure 1: An integral preference matrix (and its voting graph $D_M$) that is $2$-rationalizable using a single voter.
• Figure 2: A half-integral preference matrix (and its voting graph $D_M$) that is $2$-rationalizable using two voters.
• Figure 3: A generic preference matrix that is $3$-rationalizable using a single voter.
• Figure 4: A $2$-rationalizable matrix consistent with two voters whose partial orders are disjoint chains.
• Figure 5: A half-integral preference matrix with its voting graph and unamimity graph.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Theorem 4
• Theorem 5
• proof
• Corollary 5.1
• Theorem 6
• proof