## Abstract

In this paper, we propose a fractional preference model for the facility location game with two facilities that serve the similar purpose on a line where each agent has his location information as well as fractional preference to indicate how well they prefer the facilities. The preference for each facility is in the range of [0, L] such that the sum of the preference for all facilities is equal to 1. The utility is measured by subtracting the sum of the cost of both facilities from the total length L where the cost of facilities is defined as the multiplication of the fractional preference and the distance between the agent and the facilities.

We first show that the lower bound for the objective of minimizing total cost is at least Ω(n3). Hence, we use the utility function to analyze the agents' satification. Our objective is to place two facilities on [0, L] to maximize the social utility or the minimum utility. For each objective function, we propose deterministic strategy-proof mechanisms. For the objective of maximizing the social utility, we present an optimal deterministic strategy-proof mechanism in the case where agents can only misreport their locations. In the case where agents can only misreport their preferences, we present a 2-approximation deterministic strategy-proof mechanism. Finally, we present a 4-approximation deterministic strategy-proof mechanism and a randomized strategy-proof mechanism with an approximation ratio of 2 where agents can misreport both the preference and location information. Moreover, we also give a lower-bound of 1.06. For the objective of maximizing the minimum utility, we give a lower-bound of 1.5 and present a 2-approximation deterministic strategy-proof mechanism where agents can misreport both the preference and location.

Original language | English |
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Title of host publication | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 |

Publisher | AAAI press |

Pages | 1039-1046 |

Number of pages | 8 |

Volume | 32(1) |

ISBN (Electronic) | 9781577358008 |

DOIs | |

Publication status | Published - 25 Apr 2018 |

Externally published | Yes |

Event | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 - New Orleans, United States Duration: 2 Feb 2018 → 7 Feb 2018 |

### Publication series

Name | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 |
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### Conference

Conference | 32nd AAAI Conference on Artificial Intelligence, AAAI 2018 |
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Country/Territory | United States |

City | New Orleans |

Period | 2/02/18 → 7/02/18 |

### Bibliographical note

The work described in this paper was supported by a grant from Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11268616) and JSPS KAKENHI (Grant No. JP17H00761).## Keywords

- Algorithmic Mechanism Design
- Mechanisms without Money
- Facility Location