Matrix Rationalization via Partial Orders
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929368039686144 |
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| author | Totschnig, Agnes Vasishta, Rohit Vetta, Adrian |
| author_facet | Totschnig, Agnes Vasishta, Rohit Vetta, Adrian |
| contents | 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 $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20976 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matrix Rationalization via Partial Orders Totschnig, Agnes Vasishta, Rohit Vetta, Adrian Discrete Mathematics Computer Science and Game Theory 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 $M$. |
| title | Matrix Rationalization via Partial Orders |
| topic | Discrete Mathematics Computer Science and Game Theory |
| url | https://arxiv.org/abs/2405.20976 |