  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 $M$.
