## Abstract

Consider a set of voters V, represented by a multiset in a metric space (X,d). The voters have to reach a decision - a point in X. A choice p ? X is called a ß-plurality point for V, if for any other choice q ? X it holds that |{v ? V | ß · d(p,v) = d(q,v)}| = ^{|V}_{2}^{|}. In other words, at least half of the voters “prefer” p over q, when an extra factor of ß is taken in favor of p. For ß = 1, this is equivalent to Condorcet winner, which rarely exists. The concept of ß-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [SoCG 2020] as a relaxation of the Condorcet criterion. Denote by ß_{(}^{*}_{X,d}_{)} the value sup{ß | every finite multiset V in X admits a ß-plurality point}. The parameter ß^{*} determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane ß_{(}^{*}_{R}2_{,k·k}_{2)} = ^{v}_{2}^{3}, and more generally, for ddimensional Euclidean space, v^{1}_{d} = ß_{(}^{*}_{R}d_{,k·k}_{2)} = ^{v}_{2}^{3}. In this paper, we show that 0.557 = ß_{(}^{*}_{R}d_{,k·k}_{2)} for any dimension d (notice that v^{1}_{d} < 0.557 for any d = 4). In addition, we prove that for every metric space (X,d) it holds that v2 - 1 = ß_{(}^{*}_{X,d}), and show that there exists a metric space for which ß_{(}^{*}_{X,d}_{)} = ^{1}_{2}

Original language | English |
---|---|

Title of host publication | 35th AAAI Conference on Artificial Intelligence, AAAI 2021 |

Publisher | Association for the Advancement of Artificial Intelligence |

Chapter | Technical Tracks |

Pages | 5407-5414 |

Number of pages | 8 |

Volume | 35 |

ISBN (Electronic) | 978-171383597-4 |

ISBN (Print) | 2159-5399 |

State | Published - 2021 |

Externally published | Yes |

Event | 35th AAAI Conference on Artificial Intelligence, AAAI 2021 - Virtual, Online Duration: 2 Feb 2021 → 9 Feb 2021 |

### Publication series

Name | 35th AAAI Conference on Artificial Intelligence, AAAI 2021 |
---|---|

Volume | 6B |

### Conference

Conference | 35th AAAI Conference on Artificial Intelligence, AAAI 2021 |
---|---|

City | Virtual, Online |

Period | 2/02/21 → 9/02/21 |

### Bibliographical note

Funding Information:Funding. Work by Arnold Filtser was supported by the Si-mons Foundation. Work by Omrit Filtser was supported by the Eric and Wendy Schmidt Fund for Strategic Innovation, by the Council for Higher Education of Israel, and by Ben-Gurion University of the Negev.

Publisher Copyright:

Copyright © 2021, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.