Voting Profiles Admitting All Candidates as Knockout Winners
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917158471073792 |
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| author | De Baets, Bernard De Santis, Emilio |
| author_facet | De Baets, Bernard De Santis, Emilio |
| contents | A set of $2^n$ candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the $n$-th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For $2^n$ candidates with $n \geq 3$, we identify a voting profile with $4n -3$ voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_01382 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Voting Profiles Admitting All Candidates as Knockout Winners De Baets, Bernard De Santis, Emilio Combinatorics Probability 91B12 (Primary) 91B12, 91B14 (Secondary) A set of $2^n$ candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the $n$-th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For $2^n$ candidates with $n \geq 3$, we identify a voting profile with $4n -3$ voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large. |
| title | Voting Profiles Admitting All Candidates as Knockout Winners |
| topic | Combinatorics Probability 91B12 (Primary) 91B12, 91B14 (Secondary) |
| url | https://arxiv.org/abs/2407.01382 |