多维曼哈顿偏好
Multidimensional Manhattan Preferences
AI总结：
研究d-曼哈顿偏好谱系的存在性与极小反例，证明当d≥min(n,m-1)时所有偏好谱系均为d-曼哈顿，并刻画2-曼哈顿的极小禁止子结构。
摘要：
一个偏好谱系（即选民对一组备选方案的线性偏好序的集合）具有$m$个备选方案和$n$个选民，称为$d$-曼哈顿（相应地，$d$-欧几里得），如果备选方案和选民都可以放置在$d$维空间中，使得在每对备选方案之间，每个选民偏好与其曼哈顿（相应地，欧几里得）距离较短的方案。我们研究$d$-曼哈顿偏好谱系如何依赖于$m$和$n$的值。首先，我们提供显式构造，证明当$d \ge \min(n, m - 1)$时，每个具有$m$个备选方案和$n$个选民的偏好谱系都是$d$-曼哈顿的。我们进一步将这一积极结果推广到其他$p$-范数，其中$p \in R_{\ge 1} \cup \{\infty\}$。其次，对于$d = 2$，我们发展出禁止子结构——小规模选民集合中的偏好模式，这些模式约束任何2-曼哈顿嵌入——并利用它们证明最小的非2-曼哈顿偏好谱系要么有3个选民和6个备选方案，要么有4个选民和5个备选方案，要么有5个选民和4个备选方案。这比$d$-欧几里得偏好的情况更复杂（参见(Bogomolnaia and Laslier, 2007)和(Bulteau and Chen, 2022)）。我们还证明$d$-曼哈顿偏好蕴含$(2d-1)$维单峰性，而2-曼哈顿性与单峰性和单交叉性不可比较。
英文摘要：
A preference profile (i.e., a collection of linear preference orders of the voters over a set of alternatives) with $m$ alternatives and $n$ voters is $d$-Manhattan (resp. $d$-Euclidean) if both the alternatives and the voters can be placed into a $d$-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. We study how $d$-Manhattan preference profiles depend on the values $m$ and $n$. First, we provide explicit constructions to show that each preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan whenever $d \ge \min(n, m - 1)$. We further extend this positive result for other $p$-norms with $p \in R_{\ge 1} \cup \{\infty\}$. Second, for $d = 2$, we develop forbidden substructures-preference patterns among small sets of voters that constrain any 2-Manhattan embedding -- and use them to show that the smallest non-2-Manhattan preference profile has either 3 voters and 6 alternatives, or 4 voters and 5 alternatives, or 5 voters and 4 alternatives. This is more complex than the case with $d$-Euclidean preferences (see (Bogomolnaia and Laslier, 2007) and (Bulteau and Chen, 2022)). We also show that $d$-Manhattan preferences imply $(2d-1)$-dimensional single-peakedness, while 2-Manhattanness is incomparable with single-peakedness and single-crossingness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
