Jiehua Chen

We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters.
In particular, we show the following.

1. Every preference profile with up to two voters is one-dimensional Euclidean if and only if it is single-peaked.

2. Every preference profile with up to five alternatives is one-dimensional Euclidean if and only if it is single-peaked and single-crossing.

By the results of Chen et al. [2017], we thus obtain that the smallest single-peaked and single-crossing preference profiles that are not one-dimensional Euclidean consist of three voters and six alternatives.

We prove our second main result by describing a computer program that finds all possible preference profiles with up to five alternatives that are both single-peaked and single-crossing, and uses the publicly available CPLEX solver to provide a one-dimensional Euclidean embedding for each of these profiles. The code and the embeddings for all produced profiles are available through the code-link and the embedding-link.

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The paper can be found on arXiv.