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 $M$.

翻译：偏好矩阵$M$记录了选举中每对候选人之间的偏好关系，其元素$p_{ij}$表示偏好候选人$i$胜过候选人$j$的选民比例。若该矩阵与一组偏好为全序的选民群体相一致，则称该矩阵是可合理化的。一个著名的开放性问题要求对可合理化偏好矩阵进行简洁的表征。本文中，我们将该矩阵可合理化问题推广至更一般情形，研究偏好矩阵何时与一组偏好为宽度$\alpha$的偏序关系的选民群体相一致。偏序的宽度（即反链的最大基数）是衡量选民理性程度的自然度量；事实上，宽度为$1$的偏序即为全序。我们的研究重点聚焦于合理化数——即使得偏好矩阵可合理化的最小宽度要求。我们提出了两个主要结果。第一个结果针对半整数偏好矩阵类，证明了评估合理化数所需的关键参数是与偏好矩阵$M$相关联的无向一致性图的染色数。第二个结果针对整数偏好矩阵类，证明了此时的关键参数是与$M$相关联的有向投票图的双色数。