论文标题
多维曼哈顿偏好
Multidimensional Manhattan Preferences
论文作者
论文摘要
如果可以将替代方案和选民都放在每对替代方案之间，那么每个选民都可以选择一个较短的曼哈顿（euclidean），那么$ m $替代品和$ n $选民的优先档案是$ d $ -mmanhattan（分别为$ $ d $ -euclidean），以使每个选民之间的替代方案和选民都可以在每对替代方案之间，每个选民都有一个较短的曼哈顿（euclidean）。继Bogomolnaia和Laslier之后[数学经济学杂志，2007年]以及Chen and Grottke [社会选择和福利，2021年]，他们看着$ D $ -D $ -Euclidean的偏好概况，我们研究哪些偏好概况是$ D $ -MMANHATTAN，取决于值$ M $ M $和$ n $。 首先，我们表明，每当$ d $ d $ $ \ geq $ min（$ n $，$ m $ -1 $ 1 $）时，每个带有$ m $替代品和$ n $选民的优先档案为$ d $ -manhattan。其次，对于$ d = 2 $，我们表明，最小的非$ d $ - 曼哈顿优先化资料具有三个选民和六个替代方案，或四名选民和五个替代方案，或五名选民和四个替代方案。这比$ d $ -euclidean偏好的情况更为复杂（请参阅[Bogomolnaia和Laslier，2007年]和[Bulteau and Chen，2020]。
A preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan (resp. $d$-Euclidean) if both the alternatives and the voters can be placed into the $d$-dimensional space such that between each pair of alternatives, every voter prefers the one which has a shorter Manhattan (resp. Euclidean) distance to the voter. Following Bogomolnaia and Laslier [Journal of Mathematical Economics, 2007] and Chen and Grottke [Social Choice and Welfare, 2021] who look at $d$-Euclidean preference profiles, we study which preference profiles are $d$-Manhattan depending on the values $m$ and $n$. First, we show that each preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan whenever $d$ $\geq$ min($n$, $m$-$1$). Second, for $d = 2$, we show that the smallest non $d$-Manhattan preference profile has either three voters and six alternatives, or four voters and five alternatives, or five voters and four alternatives. This is more complex than the case with $d$-Euclidean preferences (see [Bogomolnaia and Laslier, 2007] and [Bulteau and Chen, 2020].
