The facility location game () was introduced by Procaccia and Tannenholtz . In this setting, a central planner wants to build a facility that serves agents located on a path. The agents report their locations, which are fed to a mechanism that decides where the facility should be built. Procaccia and Tannenholtz studied two different objectives that the planner seeks to minimize: the sum of the distances from the facility to all agents and the maximum distance of any agent to the facility.
Every agent aims to maximize their welfare, which increases as their distance to the facility decreases. An agent or a coalition of agents can misreport their location(s) to try to increase their welfare. It is natural to seek strategyproof (SP) or group-strategyproof (GSP) mechanisms, which incentivize truthful reporting. Often such mechanisms cannot simultaneously optimize the planner’s objective. In these cases, it is desirable to approximately optimize the planner’s objective.
In real scenarios, an agent might dislike a certain facility, such as a power plant, and want to stay away from it. This variant, called the obnoxious facility location game (), was introduced by Cheng et al., who studied the problem of building an obnoxious facility on a path . In the present paper, we consider the problem of building multiple obnoxious facilities on a path. With multiple facilities, there are different ways to define the welfare function. For example, in the case of two facilities, the welfare of the agent can be the sum, minimum, or maximum of the distances to the two facilities. In our work, as all the facilities are obnoxious, a natural choice for welfare is the minimum distance to any obnoxious facility: the closest facility to an agent causes them the most annoyance, and if it is far away, then the agent is satisfied.
A facility might not be universally obnoxious. Consider, for example, a school or sports stadium. An agent with no children might consider a school to be obnoxious due to the associated noise and traffic, while an agent with children might not consider it to be obnoxious. Another agent who is not interested in sports might similarly consider a stadium to be obnoxious. We assume that each agent has dichotomous preferences; they dislike some subset of the facilities and are indifferent to the others. Each agent reports a subset of facilities to the planner. As the dislikes are private information, the reported subset might not be the subset of facilities that the agent truly dislikes. On the other hand, we assume that the agent locations are public and cannot be misreported.
In this paper, we study a variant of , which we call (Dichotomous Obnoxious Facility Location Game), that combines the three aspects mentioned above: multiple (heterogeneous) obnoxious facilities, minimum distance as welfare, and dichotomous preferences. We seek to design mechanisms that perform well with respect to either a utilitarian or egalitarian objective. The utilitarian objective is to maximize the social welfare, that is, the total welfare of all the agents. A mechanism that maximizes social welfare is said to be efficient. The egalitarian objective is to maximize the minimum welfare of any agent. For both objectives, we seek mechanisms that are SP, or better yet, weakly or strongly group-strategyproof (WGSP / SGSP).
1.1 Our contributions
We study with agents. In Section 4, we consider the utilitarian objective. We present -approximate SGSP mechanisms for any number of facilities when the agents are located on a path, cycle, or square. We obtain the following two additional results for the path setting. In the first main result of the paper, we obtain a mechanism that is WGSP for any number of facilities and efficient for up to three facilities. To show that this mechanism is WGSP, we relate it to a weighted approval voting mechanism. To prove its efficiency, we identify two crucial properties that the welfare function satisfies, and we use an exchange argument. For the path setting, we also show that no SGSP mechanism can achieve an approximation ratio better than , even for one facility.
In Section 5, we consider the egalitarian objective. We provide SP mechanisms for any number of facilities when the agents are located on a path, cycle, or square. In the second main result of the paper, we prove that the approximation ratio achieved by any WGSP mechanism is , even for two facilities. Also, we present a straightforward -approximate WGSP mechanism. Both of the results for WGSP mechanisms hold for when the agents are located on a path or cycle. Table 1 summarizes our results.
1.2 Related work
was introduced by Procaccia and Tannenholtz . Many generalizations and extensions of have been studied [1, 8, 11, 12, 13, 14, 19, 27]; here we highlight some of the most relevant work. Cheng et al. introduced and presented a WGSP mechanism to build a single facility on a path . Later they extended the model to cycles and trees . A complete characterization of single-facility SP/WGSP mechanisms for paths has been developed [16, 17]. Duan et al. studied the problem of locating two obnoxious facilities at least distance apart . Other variants of have been considered [5, 15, 20, 25].
Agent preferences over the facilities were introduced to in  and . Serafino and Ventre studied for building two facilities where each agent likes a subset of the facilities . Anastasiadis and Deligkas extended this model to allow the agents to like, dislike, or be indifferent to the facilities . The aforementioned works address linear (sum) welfare function. Yuan et al. studied non-linear welfare functions (max and min) for building two non-obnoxious facilities ; their results have subsequently been strengthened [4, 18]. In the present paper, we initiate the study of a non-linear welfare function (min) for building multiple obnoxious facilities.
The problems considered in this paper involve a set of agents located on a path, cycle, or square. In the path (resp., cycle, square) setting, we assume without loss of generality that the path (resp., cycle, square) is the unit interval (resp., unit-circumference circle, unit square). We map the points on the unit-circumference circle to , in the natural manner. Thus, in the path (resp., cycle, square) setting, each agent is located in (resp., , ). The distance between any two points and is denoted . In the path and square settings, is defined as the Euclidean distance between and . In the cycle setting, , is defined as the length of the shorter arc between and . In all settings, we index the agents from . Each agent has a specific location in the path, cycle, or square. A location profile
is a vectorof points, where denotes the number of agents and is the location of agent . Sections 4.1 and 5.1 (resp., Sections 4.2 and 5.2, Sections 4.3 and 5.3) present our results for the path (resp., cycle, square) setting.
Consider a set of agents through and a set of facilities , where we assume that each agent dislikes (equally) certain facilities in and is indifferent to the rest. In this context, we define an aversion profile as a vector where each component is a subset of . We say that such an aversion profile is true if each component is equal to the subset of disliked by agent . In this paper, we also consider reported aversion profiles where each component is equal to the set of facilities that agent claims to dislike. Since agents can lie, a reported aversion profile need not be true. For any aversion profile and any subset of agents , (resp., ) denotes the aversion profile for the agents in (resp., not in) . For a singleton set of agents , we abbreviate as .
An instance of the dichotomous obnoxious facility location () problem is given by a tuple where denotes the number of agents, there is a set of facilities to be built, is a location profile for the agents, and is an aversion profile (true or reported) for the agents with respect to . A solution to such a instance is a vector where component specifies the point at which to build . We say that a instance is true (resp., reported) if the associated aversion profile is true (resp., reported). For any instance and any in , we define as , and as .
For any instance and any associated solution , we define the welfare of agent , denoted , as , i.e., the minimum distance from to any facility in . Remark: If is empty, we define as in the cycle setting, in the path setting, and the maximum distance from to a corner in the square setting.
The foregoing definition of agent welfare is suitable for true instances, and is only meaningful for reported instances where the associated aversion profile is close to true. In this paper, reported aversion profiles arise in the context of mechanisms that incentivize truthful reporting, so it is reasonable to expect such aversion profiles to be close to true. We define the social welfare (resp., minimum welfare) as the sum (resp., minimum) of the individual agent welfares. When the facilities are built at , the social welfare and minimum welfare are denoted by and , respectively. Thus and .
For , a algorithm is -efficient if for any instance ,
Similarly, is -egalitarian if for any instance ,
A -efficient (resp., -egalitarian) algorithm, is said to be efficient (resp., egalitarian).
We are now ready to define a -related game, which we call . It is convenient to describe a instance in terms of a pair of instances where is true and is reported. There are agents indexed from to , and a planner. There is a set of facilities to be built. The numbers and are publicly known, as is the location profile of the agents. Each component of the true aversion profile is known only to agent . Each agent submits component of the reported aversion profile to the planner. The planner, who does not have access to , runs a algorithm, call it , to map to a solution. The input-output behavior of defines a mechanism, call it ; in the special case where , we say that is a single-facility mechanism. We would like to choose so that enjoys strong game-theoretic properties. We say that is -efficient (resp., -egalitarian, efficient, egalitarian) if is -efficient (resp., -egalitarian, efficient, egalitarian). As indicated earlier, such properties (which depend on the notion of agent welfare) are only meaningful if the reported aversion profile is close to true. To encourage truthful reporting, we require our mechanisms to be SP, as defined below; we also consider the stronger properties WGSP and SGSP.
The SP property says that no agent can increase their welfare by lying about their dislikes.
A mechanism is SP if for any instance with , and , and any agent in such that , we have
The WGSP property says that if a non-empty coalition of agents lies, then at least one agent in does not increase their welfare.
A mechanism is WGSP if for any instance with , and , and any non-empty coalition such that , there exists an agent in such that
The SGSP property says that if a coalition of agents lies and some agent in increases their welfare then some agent in decreases their welfare.
A mechanism is SGSP if for any instance with , and , and any coalition such that , if there exists an agent in such that
then there exists an agent in such that
Every SGSP mechanism is WGSP and every WGSP mechanism is SP.
3 Weighted Approval Voting
Before studying efficient mechanisms for our problem, we review a variant of the approval voting mechanism . An instance of Dichotomous Voting (DV) is a tuple where voters have to elect a candidate among the set of candidates . Each voter has dichotomous preferences, that is, voter partitions all of the candidates into two equivalence classes: a top (most preferred) tier and a bottom tier . Each voter has associated (and publicly known) weights . The symbols , , and denote length- vectors with th element , , and , respectively. We now present our weighted approval voting mechanism.222Our mechanism differs from the homonymous mechanism of Massó et al., which has weights for the candidates instead of the voters .
Given a instance , every voter votes by partitioning into and . Let the weight function be such that for voter and candidate , if is in and otherwise. For any in , we define as the approval of candidate . The candidate with highest approval is declared the winner. Ties are broken according to a fixed ordering of the candidates (e.g., in favor of lower indices).
We note that the approval voting mechanism can be obtained from the weighted approval voting mechanism by setting weights to and to for all voters . In Section 2, we defined SP, WGSP, and SGSP in the setting. These definitions are easily generalized to the voting setting. Brams and Fishburn proved that the approval voting mechanism is SP . Below we prove that our weighted approval voting mechanism is WGSP (and hence also SP).
Mechanism 1 is WGSP.
Assume for the sake of contradiction that there is an instance in which a coalition of voters with true preferences all benefit by misreporting their preferences as . For any candidate , let denote the approval of when coalition reports truthfully, and let denote the approval of when coalition misreports.
Let be the winning candidate when coalition reports truthfully, and let be the winning candidate when coalition misreports. Since every voter in benefits when the coalition misreports, we know that belongs to and belongs to .
Since belongs to , we deduce that and hence . Similarly, since belongs to , we deduce that and hence .
Since wins when coalition truthfully, one of the following two cases is applicable.
Case 1: . Since and , the case condition implies that . Hence does not win when coalition misreports, a contradiction.
Case 2: and has higher priority than . Since and , the case condition implies that and has higher priority than . Hence does not win when coalition misreports, a contradiction. ∎
Mechanism 1 is not SGSP.
4 Efficient Mechanisms
4.1 Efficient mechanisms for the unit interval
We now present our efficient mechanism for .
For a given reported instance , output the lexicographically least solution in that maximizes the social welfare .
Mechanism 2 is WGSP.
Let denote a instance where and . We view each agent as a voter, and each in as a candidate. We obtain the top-tier candidates of voter , and their reported top-tier candidates , from and , respectively. Assume without loss of generality that (the other case can be handled similarly). Set and similarly . Also set and . With this notation, it is easy to see that , and that choosing the with the highest social welfare in Mechanism 2 is the same as electing the candidate with the highest approval in Mechanism 1. ∎
We show that Mechanism 2 is efficient for . First, we note a well-known result about the 1-Maxian problem. In this problem, there are points located at in the interval , and the task is to choose a point in such that the sum of the distances from that point to all s is maximized.
Lemma 1 (Optimality of the 1-Maxian Problem).
Let be a real interval, let belong to , and let denote . Then belongs to .
Let denote the reported instance, let denote the set of all in such that it is efficient to build all facilities at , and assume that is non-empty. Then is non-empty.
Let denote . When all of the facilities are built at ,
Since is non-empty, . Moreover, since does not depend on , Lemma 1 implies that
Thus, if , it is efficient to build all facilities at . Otherwise, it is efficient to build all facilities at . ∎
Mechanism 2 is efficient for .
Let denote the reported instance and let be an efficient solution for such that .
Consider fixing variables and in the social welfare function . That is, we have
For convenience, let denote and let denote for each agent .
Claim 1: For each agent , the welfare function with satisfies at least one of the following two properties:
Proof: Consider an agent . We consider five cases.
Case 1: . Since the welfare of agent is independent of the location of , is a constant function. Hence property 2 is satisfied.
Case 2: . By definition, we have . Hence property 1 is satisfied.
Case 3: . By definition, we have . Notice that . Moreover, . We consider two cases.
Case 3.1: . Then and hence for all in , that is, satisfies property 1.
Case 3.2: . Then and hence satisfies property 2.
Case 4: . This case is symmetric to Case 3 and can be handled similarly.
Case 5: . By definition, we have . Notice that . Also notice that for any in , . Hence property 1 holds.
This concludes our proof of Claim 1.
Claim 2: There is a solution that optimizes and builds facilities in at most two locations.
Proof: We establish the claim by proving that either or .
Claim 1 implies that the set of agents can be partitioned into two sets such that satisfies property 1 for all in , and satisfies property 2 for all in . Thus, we have . By Lemma 1, there is a in such that for all in . For any in , we deduce from property 2 that for all in . Therefore, for all in . This completes our proof of Claim 2.
Having established Claim 2, we can assume without loss of generality that . A similar argument as above can be used to prove that either or is an efficient solution. Now if is efficient, then one can use a similar argument to prove that either or is efficient. And if is efficient, then by applying Lemma 2 with , we deduce that either or is efficient. Thus, there is a - efficient solution. The efficiency of Mechanism 2 follows. ∎
When (resp., ), we can add one (resp., two) dummy facilities and use Theorem 4 to establish that Mechanism 2 is efficient for (resp., ). Theorem 5 below provides a lower bound on the approximation ratio of any SGSP efficient mechanism; this result implies that Mechanism 2 is not SGSP.
There is no SGSP -efficient mechanism with .
Let be a large even integer. We construct two -agent single-facility instances and . In both and , agent is located at and dislikes , agents are located at and dislike , and the remaining agents, which we denote by the set , are located at and dislike . In , all agents report truthfully, while in , all agents in report and the remaining agents report truthfully.
Let the maximum social welfare for instances and be OPT and , respectively. It is easy to see that and (obtained by building at and , respectively). Let the social welfare achieved by some SGSP mechanism on these instances be ALG and , respectively.
Let build at on . It follows that . If the agents in and agent form a coalition in and the agents in report , then the instance becomes . Thus, as is SGSP, cannot build to the right of in . Using this fact, it is easy to see that .
In view of Theorem 5, it is natural to try to determine the minimum value of for which an SGSP -efficient mechanism exists. Below we present a -efficient SGSP mechanism. It remains an interesting open problem to improve the approximation ratio of , or to establish a tighter lower bound for the approximation ratio.
Let denote the reported instance. Build all facilities at if ; otherwise, build all facilities at .
Mechanism 3 is SGSP.
Reported dislikes do not affect the locations at which the facilities are built. Hence the theorem follows. ∎
Mechanism 3 is 2-efficient.
Let denote the reported instance. Let ALG denote the social welfare obtained by Mechanism 3 on this instance, and let OPT denote the maximum possible social welfare on this instance. We need to prove that .
Assume without loss of generality that Mechanism 3 builds all facilities at . (A symmetric argument handles the case where all facilities are built at ). Then the welfare of an agent not in is and the welfare of an agent in is . Thus, . As Mechanism 3 builds the facilities at and not , we have , which implies that . Combining the above two inequalities, we have . Since no agent has welfare greater than , we have . Thus, , as required. ∎
We now establish that the analysis of Theorem 7 is tight by exhibiting a two-facility instance on which Mechanism 3 achieves half of the optimal social welfare. For the reported instance , it is easy to verify that the optimal social welfare is , while the social welfare obtained by Mechanism 3 is .
4.2 Efficient mechanism for the cycle
Now we present a simple adaptation of Mechanism 3 to the case where the agents are located on a cycle.
Let denote the reported instance. Build all facilities at if
otherwise, build all facilities at .
Mechanism 4 is SGSP.
Mechanism 4 is -efficient.
We sketch a proof that is similar to our proof of Theorem 7. Let denote the reported instance. Let ALG denote the social welfare obtained by Mechanism 4 on this instance, and let OPT denote the maximum possible social welfare on this instance. We need to prove that .
Assume without loss of generality that Mechanism 4 builds all facilities at . (A symmetric argument handles the case where all facilities are built at ). Using similar arguments, we obtain . Also, we have , and for all agents , implying that . Thus . Since no agent has welfare greater than , we have . Thus, , as required. ∎
4.3 Efficient mechanism for the unit square
We now show how to adapt Mechanism 3 to the case where the agents are located in the unit square.
Let denote the reported instance. For each point in , let denote . Let be the point in that maximizes , breaking ties lexicographically. Build all facilities at .
Mechanism 5 is SGSP.
Mechanism 5 is -efficient.
We sketch a proof that is similar to our proof of Theorem 7. Let denote the reported instance. Let ALG denote the social welfare obtained by Mechanism 5 on this instance, and let OPT denote the maximum possible social welfare on this instance. We need to prove that .
Assume without loss of generality that Mechanism 5 builds all facilities at . (A symmetric argument handles other cases). Using similar arguments, we obtain . Also, we have
for all agents , implying that . Thus . Since no agent has welfare greater than , we have . Thus, , as required. ∎
5 Egalitarian Mechanisms
We now design egalitarian mechanisms for when the agents are located on an interval, cycle, or square.
In Definition 5 below, we introduce a simple way to convert a single-facility mechanism into a mechanism. Observe that for single-facility mechanisms, specifying the input instance is equivalent to specifying .
For any single-facility mechanism , denotes the mechanism that takes as input a instance and outputs , where is the location at which builds the facility on input .
Let be a SP single-facility mechanism. Then is a SP mechanism.
Let denote a instance with , , and let be an agent such that . Let (resp., ) denote (resp., ). Since is SP, we have for each facility in . Thus , implying that agent does not benefit by reporting instead of . ∎
Let be an egalitarian single-facility mechanism. Then