On the Number of Single-Peaked Narcissistic or Single-Crossing Narcissistic Preference Profiles
Jiehua Chen, Ugo P. Finnendahl

TL;DR
This paper quantifies the number of preference profiles that are both narcissistic and either single-peaked or single-crossing, revealing exact counts for these structured voter preference profiles.
Contribution
It provides exact formulas for counting single-peaked and single-crossing narcissistic preference profiles with n voters and n alternatives.
Findings
Number of single-peaked narcissistic profiles: _{i=2}^{n-1} inom{n-1}{i-1}
Number of single-crossing narcissistic profiles: 2^{inom{n-1}{2}}
Explicit enumeration formulas for structured preference profiles.
Abstract
We investigate preference profiles for a set of voters, where each voter has a preference order on a finite set of alternatives (that is, a linear order on ) such that for each two alternatives , voter prefers to if . Such a profile is narcissistic if each alternative is preferred the most by at least one voter. It is single-peaked if there is a linear order on the alternatives such that each voter's preferences on the alternatives along the order are either strictly increasing, or strictly decreasing, or first strictly increasing and then strictly decreasing. It is single-crossing if there is a linear order on the voters such that each pair of alternatives divides the order into at most two…
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Taxonomy
TopicsGame Theory and Voting Systems · Local Government Finance and Decentralization · Economic theories and models
