Facility Location with Entrance Fees

04/24/2022
by   Mengfan Ma, et al.
Microsoft
0

In mechanism design, the facility location game is an extensively studied problem. In the classical model, the cost of each agent is her distance to the nearest facility. In this paper, we consider a new model, where there is a location-dependent entrance fee to the facility. Thus, in our model, the cost of each agent is the sum of the distance to the facility and the entrance fee of the facility. This is a refined generalization of the classical model. We study the model and design strategyproof mechanisms. For one and two facilities, we provide upper and lower bounds for the approximation ratio given by deterministic and randomized mechanisms, with respect to the utilitarian objective and the egalitarian objective. Most of our bounds are tight and these bounds are independent of the entrance fee functions. Our results are as general as possible because the entrance fee function we consider is arbitrary.

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1 Introduction

1.1 Facility location game on real line.

In the one-dimensional facility location problem, agents are located on the real line and a planner is to build facilities on the line to serve the agents. The cost of an agent is her distance to the nearest facility. The problem asks for facility locations that minimize the total cost of all agents (the utilitarian objective), or the maximum cost among all agents (the egalitarian objective). It is well-known that both optimization problems can be solved in polynomial time. Over the past decade, these problems have been intensively studied from the perspective of mechanism design. A key conversion in the models is that now each agent becomes strategic and may misreport her position to decrease her cost. These new problems are called the facility location games, which require to design mechanisms that truthfully elicit the positions of agents and (approximately) minimize the objective of total cost or maximum cost.

The seminal work of Procaccia and Tennenholtz [2009] initiates the study of mechanism design without money for the facility location game. They study strategyproof mechanisms for one and two-facility games through the lens of approximation ratio. Since then, the facility location game has become one of the main playgrounds for approximate mechanism design without money and has attracted numerous follow-up research. For the one-facility game, most results in Procaccia and Tennenholtz [2009] are already tight. For the two-facility game with the total cost objective, Lu et al. [2009] provide an upper bound of and a lower bound of for randomized strategyproof mechanisms. Lu et al. [2010] improve the lower bound to and propose a randomized strategyproof -approximation mechanism. Fotakis and Tzamos [2014] characterize deterministic strategyproof mechanisms and provide the lower bound of matching the upper bound of Procaccia and Tennenholtz [2009]. The current status of the classic facility location games on the real line is summarized in Table 1. As one can see, most bounds are tight. We note that this line of research mainly focuses on mechanisms in one and two facilities settings, and Fotakis and Tzamos [2014] have proved that, in the setting of more than two facilities, no deterministic strategyproof mechanism can achieve a bounded approximation ratio.

Variants of the classical model have also been studies flourishingly to accommodate more practical scenarios. For example, for the one-facility location game, Cheng et al. [2013] propose obnoxious facility games where every agent wants to stay away from the facility. Feigenbaum and Sethuraman [2015] and Zou and Li [2015] study the dual preference game where some agents want to stay close to the facility while the others want to stay away. Cai et al. [2016] study another criterion of fairness other than maximum cost and Li et al. [2019] study the facility game with externalities. For the two-facility location game, research has been extended from the original homogeneous facility setting to the heterogeneous facility setting where facilities serve different purposes [Serafino and Ventre 2014; Serafino and Ventre 2015; Yuan et al. 2016; Fong et al. 2018; Li et al. 2020]. There are also works setting a distance constraint between the facilities [Zou and Li 2015; Chen et al. 2018; Duan et al. 2019]. Besides, Aziz et al. [2020a] and Aziz et al. [2020b] study the location game with two capacitated facilities. The recent survey by Chan et al. [2021] depicts the state of the art.

1.2 Facility location with entrance fee.

In the classical model and its variants, the cost of an agent is measured by her distance to the closest facility. This cost can be considered as the travel fee. In innumerable real life scenarios, except for the travel fee, the agent may also need pay a service fee or entrance fee to the facility she uses, such as tickets for swimming pools and museums. The entrance fee may also differ for different positions of the facility, as the cost to build the facility on different positions may be different. Therefore, it is natural to consider entrance fees in the classical facility location problem. In this paper, we initiate the study of the facility location problem with entrance fee. In our setting, given a set of agents on the real line , the planner needs to locate a given number of facilities on . Each facility, once located, has an entrance fee determined by its location. Any agent who uses this facility is charged the entrance fee. Thus the agent incurs two costs. The first is the travel fee and the second is the entrance fee. Therefore, the overall cost of the agent using the facility is the sum of the travel fee and the entrance fee.

The facility location problem with entrance fee models many real-world scenarios. Examples are plentiful: (1) The government would like to build various social infrastructures, e.g., cinemas, hospitals, bridges, all of which charge their clients an entrance fee. (2) The voters have a single-peaked preference over the potential candidates, while after the election, the winning candidates will enforce their own tax policy on the voters. (3) The students in a classroom may need to decide the temperature of the air conditioner. The electricity bill, the amount of which depends on the temperature, is then evenly divided among the students.

For the facility location game with the entrance fee, we have a set of agents located on the real line. We need to locate a number of facilities. Each agent has her position as the private information and wants to use one facility at a minimum cost. Facility located at charges each customer the entrance fee , where is the entrance fee function. Each agent , in addition to the distance from to , also considers the entrance fee to decide whether she uses the facility. Intuitively, the agent might prefer a distant facility with a lower entrance fee than a nearby facility with a much higher entrance fee. Thus, each agent selects the facility that minimizes her overall cost. The example in Figure 1 illustrates the concept of entrance fee function.

Figure 1: An example of entrance fee function. The horizontal axis stands for , i.e., the set of all possible positions of agents. The three black dots represent three agents. The vertical axis stands for the value of entrance fee. The curve stands for the entrance fee function . If the facility is located at , the entrance fee of the facility is . The travel fee for agent at is . Assume that there is only one facility, then the cost of agent is .

1.3 Our contribution

First, we provide a new model for the facility location problem, which firstly introduces the concept of entrance fee to the classic model. The novelty is that our model enhances the classical facility location game by adding the location-dependent entrance fee function. Our model is as general as possible as no conditions are imposed on the entrance fee functions. We believe our model can brings in an idea of revisiting many existing facility location games, such as capacitated facilities [Aziz et al. 2020a], heterogeneous facilities [Li et al. 2020] and so on. Second, for one-facility and two-facility games, we design deterministic and randomized mechanisms for the utilitarian and egalitarian objectives. We also achieve lower bound results. Table 2 summarizes our mechanism design results. Most of our bounds are tight. We note that our approximation ratios (upper bounds) and lower bounds involve a parameter , which is defined as the ratio of the greatest entrance fee to the smallest one over the domain of the entrance fee function and thus . For the classical model where , we stipulate that . Comparing Tables 1 and 2, we can conclude that, on the one hand, most of our upper bound results matches the classical upper bound results by letting . This implies that our mechanisms include the classical mechanisms as a special case. On the other hand, most of our lower bound results are greater than the classical ones, which can be seen as the difficulty introduced by the entrance fee function.

In our model, the entrance fee function is part of the input and we consider arbitrary entrance fee function. This gives rise to new challenges. First of all, the classical mechanisms are no longer strategyproof. Intuitively, consider the classical median mechanism that puts the facility at the position of the median agent, although the her travel fee is her entrance fee can be large. Thus she may deviate to a position where the sum of both fees is smaller. To tackle this, we define the optimal location for each agent as the facility location that minimize her cost and design mechanisms over the space of optimal locations of agents instead of their positions. The proof of strategyproofness needs new insights such as monotonicity of the optimal location. Second, even if we can design strategyproof mechanisms for arbitrary entrance fee function, the optimal facility location for either objective now highly depends on the structure of entrance fee function. Thus it can be difficult to analyze the approximation ratio of the mechanisms, even for one-facility game. To overcome this, we build new techniques such as domination and virtual facility to bound the solution values of our mechanisms and the optimal solution and finally achieve constant approximation ratios for arbitrary entrance fee function.

Total cost Maximum cost
UB LB UB LB
Deterministic Mechanisms one-facility
(Procaccia and Tennenholtz 2009)
(Procaccia and Tennenholtz 2009)
(Procaccia and Tennenholtz 2009) (Procaccia and Tennenholtz 2009)
two-facility
(Procaccia and Tennenholtz 2009)
(Fotakis and Tzamos 2014)
Randomized Mechanisms one-facility
(Procaccia and Tennenholtz 2009)
(Procaccia and Tennenholtz 2009)
(Procaccia and Tennenholtz 2009)
(Procaccia and Tennenholtz 2009)
two-facility
(Lu et al. 2010)
(Lu et al. 2010)
(Procaccia and Tennenholtz 2009)
Table 1: Results for one-facility and two-facility location games of the classical model without entrance fee.
Total cost Maximum cost
UB LB UB LB
Deterministic Mechanisms one-facility
(Theorem 2)
(Theorem 3)
,      if ,  if (Theorem 7 and Proposition 4) ,      if ,  if (Theorem 8 and Theorem 9) (Proposition 5 and Proposition 6)
two-facility
(Proposition 3)
(Fotakis and Tzamos 2014)
Randomized Mechanisms one-facility
(Theorem 5)
if
(Theorem 6)
if (Theorem 10 and Proposition 7)
two-facility
(Proposition 3)
(Lu et al. 2010)
Table 2: Results for one-facility and two-facility location games of the new model with entrance fee, where we use to denote the ratio of the highest entrance fee to the lowest entrance fee of the entrance fee function . Theorems and propositions of our results are in bold.

The reminder of the paper is organized as follows. In Section 2, we present the formal definitions of our model and concepts. In Section 3, we define the optimal location for an position and derive local properties such as monotonicity and domination. These properties are crucial in designing and analyzing our mechanisms. Additionally, we show that the two offline optimization versions of our model can be solved in polynomial time. Section 4 and Section 5 present our main results for the one-facility game. Section 4 focuses on the objective of total cost. We design deterministic and randomized mechanisms for one-facility game, and achieve tight upper and lower bounds. Section 5 focuses on the objective of maximum cost. We achieve tight deterministic upper and lower bounds and establish a lower bound for strategyproof randomized mechanisms. In Section 6, we extend our results for one-facility game to two-facility game. Some conclusions are drawn in Section 7.

2 Model

Let be a set of agents on real line . Let be the location of agent . The location profile of all agents is . We need to put facilities on to serve the agents. Suppose we put facility at location . This gives us the location profile of the facilities .

For an agent , if it is served by a facility , then the agent bears the travel fee measured by the distance . Furthermore, each facility charges its customers an entrance fee determined by its location. Formally, there is an entrance fee function . Then, the entrance fee of facility at location is . So if an agent selects facility , the cost incurred by the agent is the sum: . For an entrance fee function , let and . The max-min ratio of , denoted by is defined as if ; if and ; else, .

Each agent always selects a facility that minimizes the sum of her travel fee and entrance fee. So, we define the cost of agent for a given facility location profile as . If there is more than one facility that minimizes the agent’s cost, we use the tie-breaking rule:

Definition 1 (Tie-breaking rule).

Select the facility with the smallest entrance fee. If there are two such facilities with equal smallest entrance fees, select the right most one.

We consider two classical objectives: the utilitarian objective and egalitarian objective. For utilitarian objective, we want to minimize the total cost of all agents that we denote by: For egalitarian objective, we want to minimize the maximum cost of all agents that is denoted by

Remark 1.

To keep our exposition as general as possible, we assume that there is an oracle that computes for a given and for any integer and , finds the minimum of in a given interval.

In the setting of mechanism design, each agent ’s location is private. The agent is required to report location which may be different from her true position. A (deterministic) mechanism outputs facility location profile according to the entrance fee function and the reported location profile :

Definition 2 (Deterministic Mehchanism).

A deterministic mechanism is a function , where is the set of all entrance fee functions .

In the classical model, a mechanism is a function from to that given a location profile of agents outputs a location profile of facilities. Definition 2 stipulates that the entrance fee functions are part of the input of our mechanisms. Thus, for our deterministic mechanism , given an entrance fee function as an input, we get a classical deterministic mechanism .

We can also define randomized mechanisms in our setting:

Definition 3 (Randomized Mehchanism).

A randomized mechanism is a function , where is the set of all entrance fee functions and

is the set of all probability distributions over

.

For an given and a randomized mechanism , if , where

is a probability distribution, the cost of an agent

is defined as the expected cost of , i.e.,

Note that the number of facilities and the entrance fee function are publicly known. Given that an agent might misreport her location to decrease her cost, it is necessary to design (group) strategyproof mechanisms.

Definition 4 (Group strategyproof).

A mechanism is group strategyproof if for any entrance fee function , any profile , any coalition with any sub-location profile , there exists an agent such that , where is obtained from by replacing the location profile of all agents in with .

In Definition 4 if only contains one agent, then definition is referred to as strategyproof. (Group) strategyproof mechanisms may not be able to achieve the optimal value of the two objectives. Therefore we use approximation ratio to evaluate the performance of the mechanism. For an entrance fee function and location profile , let and be the optimal total cost and the optimal maximum cost of the optimization problems, respectively.

Definition 5 (Approximation ratio).

For an entrance fee function and location profile , the approximation ratio of is , the approximation ratio of is defined as , and the approximation ratio of is defined as .

The approximation ratio for the maximum cost is defined in the same way by replacing with .

3 Structural Properties

3.1 Local monotonicity and optimization

Let be an entrance fee function. Given a location of an agent, let be the facility location that minimizes the cost of the agent: . If there are multiple locations minimizing the cost, we use the tie-breaking rule in Definition 1. We single out the value :

Definition 6.

In the setting above, we call the optimal location for , and the optimal entrance fee for . We call the optimal cost for the agent .

In order to find the optimal location for , the definition asks us to search the whole space of reals. This global search can be reduced to local search in the neighborhood of as shown in the next lemma.

Lemma 1.

The optimal location for an agent at is

Proof.

If the facility is placed at location , then the cost for the agent at is . Therefore any location placed at distance more that cannot be optimal. Hence, . ∎

By Remark 1, can be computed in a constant time.

Lemma 2 (Monotonicity).

For any , we have if and only if .

Proof.

By the definition of and we have

(1)
(2)

By adding (1) and (2), we get

(3)

Assume that but . If , then (3) does not hold. If , then (3) turns into equality. By (1) and (2), we get and . Then by tie-breaking rule in Definition 1, we have . ∎

Let be the open interval between and . The next lemma connects and .

Lemma 3.

Either or .

Proof.

Let . Then by the Monotonicity Lemma. Suppose for contradiction that and . Then we have . Then

   .

This implies . Strict inequality cannot happen. Therefore, . By the tie-breaking rule in Definition 1, . ∎

Lemma 4.

Assume that . Then we have . Similarly, .

Proof.

It is clear that . Also, by the Monotonicity Lemma, we have . If or , then this lemma is clearly true. Thus, we assume that . Now consider the case when , and . Then, we have by definition. Hence,

The other cases are verified similarly. ∎

Next, we capture the property that one location is preferred by each agent over another location.

Definition 7 (Domination).

Location dominates location if for all .

Lemma 5.

For any , dominates all locations in .

Proof.

Assume without loss of generality that . Let and . For each agent , there are three cases.

Case 1: . Note that as . Therefore, we have

Case 2: . We have the following calculations:

Case 3: . Similarly: . ∎

3.2 Global optimization

By the Monotonicity Lemma, . For , let be the location that minimizes the total cost, i.e., . If there are multiple solutions, then we use the tie-breaking rule from Definition 1. Define as the location that minimizes the maximum cost similarly.

Lemma 6.

and .

Proof.

Suppose for contradiction that or . We will only disprove since can be disproved by symmetry.

Case 1: . We have . Thus dominates . This implies . This is a contradiction.

Case 2: . Then we must have because of Lemma 5. Thus, dominates . Given the tie-breaking rule in Definition 1, this is a contradiction.

In a similar way, we can prove . ∎

We would like to design effective algorithms that find the optimal facility locations.

Lemma 7.

When there is only one facility, the two optimization problems can be solved in time.

Proof.

Let be the location in such that the total cost is minimized:

By Lemma 6, it is sufficient to optimize over at most intervals and we get . To obtain , we only need to optimize over agents and . ∎

When , we have the following lemma that describes the structure of a solution.

Lemma 8.

For a given facility location profile, if two agents select the same facility, then any agent located between these two agents also selects the same facility.

Proof.

Let be the facility location profile, be three agents such that and select the same facility, say . Let be a large positive, say . Then we construct a new entrance fee function such that if for a certain and if for any . We claim that, the optimal location for an agent under is the location of the facility selected by the agent. Denote by be the optimal location for agent under entrance fee function . Thus we have . Since , by the Monotonicity Lemma, we have . ∎

Equipped with Lemmas 6 and 8, the next lemma shows that the optimal solutions when can be found in polynomial time.

Proposition 1.

For general number of facilities, the two optimization problems can be solved in time.

Proof.

Let be the set of customers of facility , be the location of facility , then a solution is denoted by where and the sets are pairwise disjoint. Recall that , we say a subset of agents is continuous if for some . By Lemma 8, we have that is continuous. Denote by the continuous set of agents .

First, we consider the problem of minimizing the total cost. We use to represent the minimum total cost when agents are divided into partitions, and are in the th partition. Thus the optimal total cost to our problem is . Denote by the optimal total cost incurred by agents when they are served by a single facility. By Lemma 7, can be computed in time.

We will make use of a array , whose entries are initially set to empty. We invoke Algorithm 1 to compute and recover the solution from the values stored in .

1:  if  or or  then
2:     Return .
3:  else if  not empty then
4:     Return .
5:  else
6:      Return
7:  end if
Algorithm 1

We argue for the correctness. The base case is clear. The total cost is zero if there are no agents or no facilities. Suppose we want to compute and we have already computed where or or . We distinguish between two cases depending on whether or not agent is still in the th partition. If agent is also in the th partition, then . Otherwise we know that is exactly on the right boundary of the th partition and agents in are served by a single facility, thus . Actually, we derive the following transition function:

It easy to see the running time of Algorithm 1 is by using memorization.

Next, we consider the problem of minimizing the maximum cost. Let be the minimum maximum cost of agents when agents are divided into partitions, and are in the th partition. Let the optimal maximum cost of agents when they are served by a single facility. In slightly different way, we derive the transition function for the maximum cost as follows, and the algorithm to compute is similar to Algorithm 1

4 Utilitarian Version With One Facility

In this section, we consider the objective of minimizing the total cost for the one-facility game. We will first investigate deterministic mechanisms, and then turn our attention to randomized mechanisms.

4.1 Deterministic Mechanisms

Let be the reported location profile by the agents. We assume that the agents are ordered such that . Denote by the mechanism that outputs the optimal location for agent for any input and .

Theorem 1.

The mechanism is group strategyproof.

Proof.

Obviously, any agent at has no incentive to misreport. To modify the output of , at least one agent located in has to misreport a location strictly greater than or at least one agent in has to misreport a location strictly smaller than . We will prove that in either case the agent will not benefit from misreporting.

We only consider the case when agent with misreports a location . The other case is proved in a similar way. Let be the th order statics in . Then we have . Let and . By Lemma 4, we have . Therefore, agent has no incentive to misreport. ∎

Let be the mechanism that always outputs the optimal location for the median agent .

We analyze the approximation ratio of for the total cost. Let , and . To ease our analysis, we introduce the notion of virtual total cost , which is the total cost of when a virtual facility is located at with entrance fee . It is possible . When , we have . The virtual cost of agent is . Clearly, when .

Observation 1.

Let , and such that . If or , then .

Next, we show that the approximation ratio of can be bounded by a constant.

Theorem 2.

For each entrance fee function with max-min ratio , the approximation ratio of for the total cost is at most . This also implies that the approximation ratio of  for the total cost is at most  .

Proof.

Assume without loss of generality that . Recall that is the optimal cost of the median agent. For any agent location , if , it is easy to see that . If , we have . Thus

(4)

Next we prove . Suppose for contradiction that , then by Observation 1 and (4) we have

This is a contradiction. Hence, there is an location between and such that

(5)

By Observation 1 we have

(6)

Thus by (6) and (4) we have

(7)

We will identify a location profile for each such that the bound given by (7) for is no greater than that for . Let and . Thus . Let be the profile such that agents are in and agents are in . By (5) we have . Let . By (7) we have

(8)

Since and , (8) is maximized when and , we have