Facility location games with optional preference
Introduction
In the classical facility location problem, a set of potential sites need to be determined to establish facilities in order to fulfill customer demands. This model has been used in supply chain design when the objective is to reduce the traveling cost between the locations of the facilities and the customers. Later on, the problem is studied in the field of game theory with the consideration of the self-interested customers (or agents in the following) who may affect the outcome of the construction sites of the facilities by reporting false information (i.e. misreporting) and hence benefit from the outcome. The problem is referred to as facility location games. In facility location games, the focus is to design a (public) mechanism to determine the construction sites of the facilities such that no customer could benefit by misreporting. The information reported by the customer is called preference, and the decision of the construction sites of the facilities relies on the reported preferences from the agents.
In this paper, we study facility location games with two facilities. The decision has to be made by the government to build two public facilities on a street where a set of self-interested agents who want to minimize their own costs are situated. Agents are required to report their private information, which is a subset of facilities that the agent prefers. The locations of the agents are assumed to be public in our model. We aim to design mechanisms in which no agent can benefit from misreporting his/her true information, and at the same time we want to optimize certain objectives such as minimizing the social cost or the maximum cost, where the cost of an agent is a function of the distances to the facilities that the agent prefers. Previous works on facility location games can be mainly classified into three categories.
1. Only one facility is built and agents may like or dislike the facility.
2. Two or more homogeneous facilities are built and agents care for the closer one.
3. Two heterogeneous facilities are built and agents' utilities are the sum of their utilities towards these two facilities.
Preference is a basic and key element in the facility location problem. For the two heterogeneous facility location game, there are three kinds of preferences, namely three kinds of agents: agents who like facility 1 (), agents who like facility 2 () and agents who like both facility 1 and facility 2 (). If an agent likes both facilities, Serafino and Ventre studied the case where the agent needs to access both facilities at the same time and therefore the cost of the agent is the sum of the distances from these two facilities [1]. In this paper, we generalize the cost function to be a non-linear function of the agent's distances to the two facilities if the agent likes both facilities. A special case of this generalization is the optional preference model we considered in this paper, where the agent's cost depends on the distance to the closer/farther facility. The optional preference model for the two heterogeneous facility location game is also a natural extension of the two homogeneous facility location game.1
We consider two variants of optional preferences: Min and Max, in which the agents preferring both facilities only care for the closer one (Minimum Distance) or the farther one (Maximum Distance) respectively. Both variants can find applications in many real life scenarios. For the Min variant, consider a local government building bus stops for two bus routes on the street. Residents on this street whose destinations for commuting are covered by two bus routes have a choice to use either of the bus routes, and to reduce their walking distance they would definitely be willing to pick the route with the closer bus stop connecting to their living place. For the rest who only have one connection option, they have to go to the respective bus stop. For the Max variant, consider a factory requiring two different raw materials from two storage sites for manufacturing. The factory has multiple trucks which can be sent out simultaneously to retrieve materials. Assuming that trucks have the same speed, the time delay for manufacturing depends on the distance to the farther site. We allow the two facilities to be put in the same location which is well justified in the bus stop scenario since the government has the option to build one bus stop for two bus routes and also in the factory scenario since the storage sites of the raw materials can be the same place.
In the scenarios mentioned above, we assume all agents know the mechanisms that the government will adopt. An agent may have a chance to reduce his/her cost by misreporting. A mechanism is strategyproof if it can guarantee that an agent cannot reduce his/her cost by misreporting (formally defined in Section 2). In addition, we need to evaluate the mechanisms in terms of optimization of the social cost (the sum of costs of all agents) or the maximum cost (the maximum cost of all agents). The evaluation is mainly conducted by the approximation ratio for the social/maximum cost of a mechanism, which is the worst ratio between the social/maximum cost of the mechanism output and the optimal social/maximum cost among all possible profiles.
Our contribution: We initiate the study of the optional preference model under two objectives: minimizing the maximum cost and minimizing the social cost. We propose strategyproof mechanisms and derive lower bounds for this new model, with results shown in Table 1.
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For the variant of Minimum Distance, to minimize the maximum cost we propose a deterministic strategyproof 2-approximation mechanism and show that it is impossible to achieve an approximation ratio better than 4/3. To minimize the social cost, we give a (+1)-approximation deterministic strategyproof mechanism where n is the number of agents and prove a lower bound of 2 for the approximation ratio of deterministic strategyproof mechanisms. We also give a lower bound of 3/2 for the approximation ratio of randomized strategyproof mechanisms.
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For the variant of Maximum Distance, we propose an optimal deterministic strategyproof mechanism to minimize the maximum cost. Meanwhile, we also propose a 2-approximation deterministic strategyproof mechanism to minimize the social cost.
Meanwhile, Cheng et al. [23] initiated the mechanism design for obnoxious facility location games, where agents have the preference to stay as far away as possible from the facility, with both deterministic and randomized group strategyproof mechanisms. Later they further extended the model into trees and circles [24]. Ye et al. [25] considered the problem with the objective of maximizing sum of squares of distances and sum of distances. They gave lower bounds and proposed both deterministic and randomized mechanisms. The dual preference model was introduced in [26], [27], where some agents want to be close to the facility while others want to be far away from the facility. This model is extended to multiple facilities by Anastasiadis and Deligkas [28] and they proved that for heterogeneous k-facility location games on a real line segment, there is no optimal deterministic or randomized strategyproof mechanism for . For two-facility location games, Zou and Li [26] also studied another model where two facilities need to be built within a certain distance and agents have choice to be close to or far away from one of the two facilities. This work is followed by Chen et al. [29] in which both location and preference are considered as private information and they proposed randomized/deterministic group strategyproof mechanisms with provable approximation ratios. Later on, Duan et al. [30] considered the additional requirement of a minimum distance between the two facilities and proposed strategyproof mechanisms. They also studied triple-preference where each of the two facilities may be favorable, obnoxious, indifferent for any agent.
There is another line of research focusing on the social choice aspect of facility location games [31], [32], [33], [34]. In that setting, there is a finite number of alternative positions to build the facility and the agents will report their preference order over those positions instead of releasing their actual private positions. The performance of the mechanism is measured by distortion, which is the worst case ratio (over all consistent utility profiles) of the social welfare of the optimal alternative to that of the alternative selected by the mechanism. Babaioff et al. [35] introduced a middle level mediators which first collect information from strategic agents and then interact with the government strategically. Each mediator aims to optimize the combined utilities of his agents and the government aims to optimize the combined utilities of all agents. They showed that when both agents and mediators act strategically, there is no strategyproof mechanism that can achieve any approximation. On the positive side, when only one level acts strategically, good approximation is possible to achieve strategyproofness.
Our work is most closely related to the work of Serafino and Ventre [1] where the locations of agents are public and the preferences of agents are considered as private information. In their work, they considered the scenario where agents and facilities are located at integer points (evenly distributed, in other words) on a straight line and made the assumption that two facilities cannot be built at the same position. They propose a mechanism, named Two Extreme mechanism, with 3-approximation for maximum cost objective where the cost of an agent is defined to be the sum of the distances to the facilities when the agent prefers both facilities. The Two Extreme mechanism places the first facility (resp. second facility) at the location of the agent who prefers it as left (resp. right) as possible, which is firstly adopted by Procaccia and Tennenholtz [2]. In our model, we further investigate the scenario in which agents are located at any point in the line, and we consider non-linear cost functions, min function and max function, i.e. the cost of an agent is defined to be the minimum or maximum of the distances to the facilities that the agent prefers.
In fact, in our model if the cost function is defined to be the sum of the distances, then the problem can be solved by handling the two facilities separately. In other words, the solution can be easily constructed based on the solutions from two single facility location games, maintaining both optimality and strategyproofness.
The remainder of the paper is organized as follows. In Section 2, we give the formulation of the problem. We consider the variant of Minimum Distance in Section 3 and the variant of Maximum Distance in Section 4. In Section 3.1, we focus on the objective of minimizing the maximum cost, while we study the social cost objective in Section 3.2 in which randomized mechanisms are also studied. For the variant of Maximum Distance, we consider the maximum cost objective in Section 4.1 and the social cost objective in Section 4.2. Finally, we summarize our results in Section 5 and discuss the connection between our model and previous work by Serafino and Ventre [1].
A preliminary version of this work appeared in the Proceedings of the Twenty-second European Conference on Artificial Intelligence (ECAI) [36]. In this current version, we further show a lower bound of 3/2 on approximation ratios for any randomized strategyproof mechanisms for the variant of Minimum Distance with the social cost objective. Moreover, we extensively simplified the analysis in Theorem 2 and Theorem 5.
Section snippets
Problem description
Given a set of agents located on a straight line, we aim to design a mechanism to build a set of facilities on a straight line. The input of the problem consists of a location profile and a preference profile , where for each agent , is the location of agent i and is the preference reported by agent i with and . A mechanism M takes location profile x and preference profile p as input and returns as output a
Minimum distance
In this section, we consider the Minimum Distance cost function, i.e. . We study maximum cost objective and social cost objective respectively and propose strategyproof mechanisms and give lower bounds for approximation ratios of strategyproof mechanisms.
Maximum distance
In this section, we consider the Maximum Distance cost function, i.e. . We study the maximum cost objective and the social cost objective respectively and propose strategyproof mechanisms.
For facility with a specified preference profile, let be the set of agents whose preference contains facility . Note that is exactly the set of agents with optional preference . Correspondingly, let be the collection of
Conclusion and discussion
We study heterogeneous facility location games with optional preference for two facilities with Minimum/Maximum Distance cost function of agents, in which an agent prefers a subset of facilities. This is a new model which covers more real life scenarios. Results are summarized in Table 1 where we have proposed several deterministic strategyproof mechanisms under different scenarios with provable approximation ratios.
For the result of Minimum Distance with social cost objective, there is a huge
Declaration of Competing Interest
None
Acknowledgements
We would like to thank the anonymous reviewers for their pertinent and insightful comments. This research was partially supported by the National Natural Science Foundation of China Grants No. 12071460, No. 61832003, 61761136014, 61872334, the 973 Program of China Grant No. 2016YFB1000201 and K. C. Wong Education Foundation. Minming Li is also from City University of Hong Kong Shenzhen Research Institute, Shenzhen, P. R. China. The work described in this paper was supported by a grant from
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