Abstract
An electorate with fully-ranked innate preferences casts approval votes over a finite set of alternatives. As a result, only partial information about the true preferences is revealed to the voting authorities. In an effort to understand the nature of the true preferences given only partial information, one might ask whether the unknown innate preferences could possibly be single-crossing. The existence of a polynomial time algorithm to determine this has been asked as an outstanding problem in the works of Elkind and Lackner [18]. We hereby give a polynomial time algorithm determining a single-crossing collection of fully-ranked preferences that could have induced the elicited approval ballots, or reporting the nonexistence thereof. Moreover, we consider the problem of identifying negative instances with a set of forbidden sub-ballots, showing that any such characterization requires infinitely many forbidden configurations.
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Notes
- 1.
This is perhaps the main open question in the literature on proportional representation [26].
- 2.
Recently, we learned that a solution has in fact been proposed as early as 1979 in the context of the simple plant location problem, by Beresnev and Davydov [4]. This paper is only available in Russian, and Russian-speaking experts seem to believe that the paper is likely missing steps in the arguments. Beresnev and Davydov [4] is referenced in [24], but without details.
- 3.
For general partial orders, they show that PSC implies SSC, but not conversely.
- 4.
This is used to prove the hardness of our problem for general weak orders, but the argument requires an unbounded number of indifference classes, so it does not work for bounded k..
- 5.
If we replace our undirected graph by the directed implications graph and the word “connected” by “strongly-connected.”.
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Constantinescu, A., Wattenhofer, R. (2024). Recovering Single-Crossing Preferences from Approval Ballots. In: Garg, J., Klimm, M., Kong, Y. (eds) Web and Internet Economics. WINE 2023. Lecture Notes in Computer Science, vol 14413. Springer, Cham. https://doi.org/10.1007/978-3-031-48974-7_11
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