Strategyproof Facility Location for Concave Cost FunctionsThis research was supported by the project AlgoNow, co-financed by the European Union (European Social Fund - ESF) and Greek national funds, through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES, investing in knowledge society through the European Social Fund.

# Strategyproof Facility Location for Concave Cost Functions††thanks: This research was supported by the project AlgoNow, co-financed by the European Union (European Social Fund - ESF) and Greek national funds, through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES, investing in knowledge society through the European Social Fund.

Dimitris Fotakis School of Electrical and Computer Engineering,
National Technical University of Athens, 157 80 Athens, Greece.
Email: fotakis@cs.ntua.gr
Christos Tzamos Computer Science and Artificial Intelligence Laboratory,
Massachusetts Institute of Technology, Cambridge, MA 02139.
Email: tzamos@mit.edu
###### Abstract

We consider -Facility Location games, where strategic agents report their locations on the real line, and a mechanism maps them to facilities. Each agent seeks to minimize his connection cost, given by a nonnegative increasing function of his distance to the nearest facility. Departing from previous work, that mostly considers the identity cost function, we are interested in mechanisms without payments that are (group) strategyproof for any given cost function, and achieve a good approximation ratio for the social cost and/or the maximum cost of the agents.

We present a randomized mechanism, called Equal Cost, which is group strategyproof and achieves a bounded approximation ratio for all and , for any given concave cost function. The approximation ratio is at most for Max Cost and at most for Social Cost. To the best of our knowledge, this is the first mechanism with a bounded approximation ratio for instances with facilities and any number of agents. Our result implies an interesting separation between deterministic mechanisms, whose approximation ratio for Max Cost jumps from to unbounded when increases from to , and randomized mechanisms, whose approximation ratio remains at most for all . On the negative side, we exclude the possibility of a mechanism with the properties of Equal Cost for strictly convex cost functions. We also present a randomized mechanism, called Pick the Loser, which applies to instances with facilities and only agents. For any given concave cost function, Pick the Loser is strongly group strategyproof and achieves an approximation ratio of for Social Cost.

Keywords: Algorithmic Mechanism Design; Social Choice; Facility Location Games

## 1 Introduction

We consider -Facility Location games, where facilities are placed on the real line based on the preferences of strategic agents. Such problems are motivated by natural scenarios in Social Choice, where the government plans to build a fixed number of public facilities in an area (see e.g., [12]). The choice of the locations is based on the preferences of local people, or agents. Each agent reports his ideal location, and the government applies a (deterministic or randomized) mechanism that maps the agents’ preferences to facility locations.

The agents evaluate the outcome of the mechanism according to their connection cost, given by a nonnegative increasing function of the distance of their ideal location to the nearest facility. Agents seek to minimize their connection cost, and may misreport their ideal locations in an attempt of manipulating the mechanism. Therefore, the mechanism should be strategyproof, i.e., should ensure that no agent can benefit from misreporting his location, or even group strategyproof, i.e., should ensure that for any coalition of agents misreporting their locations, at least one of them does not benefit. The government’s goal is to minimize an objective function of the agents’ connection cost. Most prominent among them are the objective of Social Cost, which considers the total cost of the agents, and the objective of Max Cost, which considers the maximum cost of an agent. So, in addition to (group) strategyproofness, the mechanism should either optimize or achieve a reasonable approximation to the designated objective function, thus ensuring that the outcome is socially efficient.

Previous Work. The numerous applications and the elegance of the model have attracted a significant volume of research on the problem. In Social Choice, the emphasis has been on characterizing the class of (group) strategyproof mechanisms for locating a single facility if the agents’ preferences are single-peaked. Roughly speaking, an agent has single-peaked preferences if he has an ideal location (or peak), and consistently prefers less the locations farther from it. However, the strength of his preference for locations closer to his peak is not explicitly quantified by any function of the distance. For general single-peaked preferences, a classical result of Moulin [13] shows that the class of deterministic strategyproof mechanisms for locating a single facility on the line coincides with the class of generalized median mechanisms (see also the surveys of Barberá [2] and Sprumont [18], and [14, Chapter 10]). Schummer and Vohra [17] extended this characterization to tree metrics, and proved that for non-tree metrics, any onto strategyproof mechanism must be a dictatorship. More recently, Dokow et al. [3] obtained similar characterizations for locating a single facility on the discrete line and on the discrete circle.

Adopting an optimization viewpoint to Facility Location games, Procaccia and Tennenholtz [16] introduced the framework of approximate mechanism design without money. The basic idea is to consider game-theoretic versions of optimization problems, such as -Facility Location, where efficiency is quantified by an objective function (instead of efficiency related properties, such as onto, non-dictatorship, and Pareto-efficiency, typically studied in Social Choice). Then, any reasonable approximation to the optimal solution can be regarded as a socially desirable outcome, and one seeks to determine the best approximation ratio achievable by strategyproof mechanisms. As for the preferences of the agents, with respect to which strategyproofness is defined, this line of research adopted the standard definition of Facility Location problems from Operations Research (see e.g., [11]). Thus, it implicitly abandoned the setting of general single-peaked preferences, in favor of the more restricted (and technically easier to handle) case where the agents’ cost is given by a linear function of their distance to the nearest facility. Translated into this framework, the results of [13, 17] imply a deterministic strategyproof mechanism that minimizes the Social Cost for 1-Facility Location on the line and in tree metrics. On the negative side, the impossibility result of [17] implies that the best approximation ratio achievable for the objective of Social Cost by deterministic strategyproof mechanisms for 1-Facility Location in general metrics is . However, the explicit quantification of agents’ preferences now allows for randomized mechanisms that are strategyproof with respect to the agents’ expected cost (a.k.a. incentive compatible in expectation, see e.g., [14, Section 9.5.6]) and may achieve better approximation ratios.

Since [16], there has been a considerable interest in quantifying the best approximation ratio achievable by strategyproof mechanisms for -Facility Location on the line and in general metric spaces. As a result, the approximability of -Facility Location (with linear cost functions) by deterministic and randomized strategyproof mechanisms has become well understood in many interesting cases (see also Fig. 1). The main message is that deterministic strategyproof mechanisms can only achieve a bounded approximation ratio if we have at most facilities [16, 6]. On the other hand, randomized mechanisms achieve better approximation ratios for -Facility Location, and also a bounded approximation ratio if we have facilities and only agents [4]. Notably, such instances are known to be hard for deterministic mechanisms. In particular, the inapproximability of -Facility Location by anonymous deterministic strategyproof mechanisms, for all , was proved in [6] for instances with only agents.

Motivation and Contribution. Our work is motivated by two natural questions related to approximate mechanism design without money for -Facility Location. The first question is about the approximability of -Facility Location by randomized strategyproof mechanisms for instances with any number of facilities and any number of agents. Prior to this work, we have only known randomized mechanisms with a bounded111The approximation ratio of a mechanism for -Facility Location is bounded if it is a function of and . We highlight that this property is essentially objective-independent, since any mechanism with a bounded approximation ratio for e.g., Max Cost also has a bounded approximation for Social Cost and for the objective of minimizing the norm of the agents’ costs, for any , and vice versa. approximation ratio if we have either at most 3 facilities or facilities and only agents. Most importantly, all the randomized upper bounds in Fig. 1 are achieved by mechanisms that balance between strategyproofness and efficiency using different approaches (see e.g., [16, 9, 4]).

The second question is whether the restriction to linear cost functions is a necessary price to pay for adopting the elegant optimization framework of Procaccia and Tennenholtz [16] and aiming at a reasonable approximation ratio. In fact, we can imagine a few natural scenarios where the agents’ cost is best described by a convex or a concave non-decreasing cost function of their distance to the nearest facility. For example, a convex cost function captures the fact that the growth rate of the people’s disutility from commuting increases with the distance (e.g., in addition to cost and time considerations, people get more and more tired if they commute over long distances). On the other hand, a concave cost function captures the fact that the growth rate of the traveling time decreases with the distance (e.g., people walk over short distances, bike over medium distances, drive over long distances, and take a plane over really long ones). To a certain extent, a setting where the agents’ cost function is not fixed, but is given as part of the input, would be closer to the setting of general single-peaked preferences in Social Choice. Then, a mechanism should be strategyproof, or even group strategyproof, for any given cost function , just as generalized median mechanisms are strategyproof for any collection of single-peaked preferences, while the approximation ratio may also depend on some quantitative properties (e.g., the derivative) of . Notably, this holds for the class of percentile mechanisms [19], which decide on the facility locations based on the ordering of the agents on the line, are group strategyproof, and include the optimal (wrt. the approximation ratio for linear cost functions) deterministic mechanisms for and -Facility Location on the line. However, percentile mechanisms have an unbounded approximation ratio for all . In contrast, the strategyproofness of known randomized mechanisms crucially depends on the linearity of the cost function (see e.g., [16, Mechanism 1] which is not strategyproof e.g., for ).

In this work, we make significant progress in both research directions above. Our main technical contribution consists of two randomized mechanisms, called Equal Cost and Pick the Loser, that are group strategyproof and achieve a bounded approximation ratio for any number of facilities and any given concave cost function.

Equal Cost, presented in Section 3, applies to instances with any number of facilities and any number of agents , and is the first (group) strategyproof mechanism with a bounded approximation ratio for all and . Its approximation ratio is at most for Max Cost and at most for Social Cost, for all concave cost functions . Combined with the lower bound of [16] for the objective of Max Cost, this implies that the best approximation ratio achievable by randomized mechanisms for -Facility Location on the line and is at least and at most , for all and for all concave cost functions. Moreover, we obtain an interesting separation between deterministic mechanisms, whose approximation ratio for Max Cost jumps from to unbounded when increases from to , and randomized mechanisms, whose approximation ratio remains a small constant for all .

From a technical viewpoint, Equal Cost works by equalizing the expected cost of all agents. The mechanism first covers the agents’ locations with disjoint intervals of length , where is chosen so that is at most twice the optimal maximum cost of an agent. Then, taking the cost function into account, it computes a random variable in , so that all locations have the same expected cost, under , if is connected to a facility distributed in according to . Finally, Equal Cost places a facility in each interval according to the random variable so that all agents have an expected cost equal to the expectation of .

The key technical claim in the analysis of Equal Cost is that if the cost function is concave and piecewise linear, a random variable with the desired properties exists and can be computed efficiently as the solution to a homogeneous system of linear equations (Lemma 2). This claim can be generalized to any continuous concave function, but the technical details have to do with techniques for the solution of integral equations and are beyond the scope of this work. We show that Equal Cost is (resp. strongly) group strategyproof for any given (resp. strictly) concave cost function , and that the agents’ expected cost is at most the maximum cost of an agent in the optimal solution for the objective of Max Cost (Lemma 5). In addition to implying the approximation guarantees, the upper bound on the expected cost of the agents indicates that the facility allocation of Equal Cost is fair in expectation, and does not unnecessarily increase the agents’ disutility.

To demonstrate the natural behavior of Equal Cost for typical cost functions, we derive the exact form of the random variable for three important cases: linear cost functions, piecewise linear cost functions with two pieces, and exponential cost functions of the form (Section 4). Moreover, we show how to implement Equal Cost if the agents and the facilities should lie in a bounded interval (Section 5). This implies that Equal Cost can be applied to instances where the agents lie on a circle metric, with the same approximation guarantees, but rather surprisingly, with group strategyproofness carrying over only if the number of facilities is even.

On the negative side, we exclude the possibility of a mechanism with the properties of Equal Cost for strictly convex cost functions (Section 5.2). Specifically, we show that the expected cost of the agents in the same interval cannot be equalized if the cost function is strictly convex. Moreover, employing an exponential cost function, we show (Lemma 9) that there does not exist a randomized strategyproof mechanism with a bounded approximation ratio for any given convex cost function (note that the approximation ratio here may also depend on the cost function).

In Section 6, we focus on the simpler and elegant setting where we have facilities and only agents. This setting was motivated and studied in [4], and deserves special attention not only because such instances are among the hardest ones for deterministic mechanisms (see e.g., [6, Theorem 7.1]), but also because they make Equal Cost perform poorly for the objective of Social Cost. We present the Pick the Loser mechanism that allocates facilities to all but a single agent, designated as the loser. The probability distribution according to which the loser is chosen is motivated by the probability distribution used by [8] for scheduling on selfish unrelated machines. Our key technical contribution here is to show that Pick the Loser is strongly group strategyproof for any given concave cost function (Lemma 10). We also show that Pick the Loser achieves an approximation ratio of for the objective of Social Cost. Thus, we significantly improve on the previously best known approximation ratio of achieved by the Inversely Proportional mechanism of [4] for this class of instances. Moreover, the small approximation ratio of Pick the Loser nicely complements the poor performance of Equal Cost for such instances.

Other Related Work. For the objective of Max Cost, Alon et al. [1] almost completely characterized the approximation ratios achievable by randomized and deterministic mechanisms for 1-Facility Location in general metrics and rings. For the objective of Social Cost, Nissim et al. [15] and Fotakis and Tzamos [7] considered imposing randomized mechanisms that achieve an additive approximation of and an approximation ratio of for -Facility Location on the line and in general metric spaces, respectively. For -Facility Location on the line and the objective of minimizing the norm of the agents’ distances to the facility, Feldman and Wilf [5] proved that the best approximation ratio is for randomized and for deterministic mechanisms. Moreover, they presented a class of randomized mechanisms that includes all known strategyproof mechanisms for 1-Facility Location on the line.

## 2 Notation, Definitions, and Preliminaries

For a random variable , we let denote the expectation of . For an event in a sample space, we let denote the probability that occurs.

Instances. We consider -Facility Location with facilities and agents on the real line. We let be the set of agents. Each agent resides at a location , which is ’s private information. An instance is a tuple , where is the agents’ locations profile and is a cost function that gives the connection cost of each agent. The cost function is public knowledge and the same for all agents. Normalizing , we assume that . If the cost function is clear from the context, we let an instance simply consist of .

For an -tuple , we let be without . For a non-empty set of indices, we let and . We write to denote the tuple with in place of , to denote the tuple with in place of and in place of , and so on.

Mechanisms. A deterministic mechanism for -Facility Location maps an instance to a -tuple , , of facility locations. We let (or simply , whenever is clear from the context) denote the outcome of for instance , and let denote , i.e., the -th smallest coordinate in . We write to denote that has a facility at location . A randomized mechanism maps an instance to a probability distribution over -tuples .

Connection Cost, Social Cost, Maximum Cost. Given a -tuple , , of facility locations, the connection cost of agent with respect to , denoted , is . Given a deterministic mechanism and an instance , we let (or simply, , if is clear from the context) denote the connection cost of agent with respect to the outcome of . If is a randomized mechanism, the expected connection cost of agent is

 cost(xi,F(→x,c))=IE→y∼F(→x,c)[cost(xi,→y)]

The Max Cost of a deterministic mechanism for an instance is

 MC[F(→x,c)]=maxi∈Ncost(xi,F(→x,c))

The expected Max Cost of a randomized mechanism for an instance is

 MC[F(→x,c)]=IE→y∼F(→x,c)[maxi∈Ncost(xi,→y)]

The optimal Max Cost, denoted , is .

The (resp. expected) Social Cost of a deterministic (resp. randomized) mechanism for an instance is . The optimal Social Cost, denoted , is .

Approximation Ratio. A (randomized) mechanism for -Facility Location achieves an approximation ratio of for a class of cost functions and the objective of Max Cost (resp. Social Cost), if for all cost functions and all location profiles , (resp.  ).

Strategyproofness and Group Strategyproofness. A mechanism is strategyproof for a class of cost functions if no agent can benefit from misreporting his location. Formally, is strategyproof if for all cost functions , all location profiles , any agent , and all locations ,

 cost(xi,F(→x,c))≤cost(xi,F((→x−i,y),c)).

A mechanism is (weakly) group strategyproof for a class of cost functions if for any coalition of agents misreporting their locations, at least one of them does not benefit. Formally, is (weakly) group strategyproof if for all cost functions , all location profiles , any non-empty coalition , and all location profiles for , there exists some agent such that

 cost(xi,F(→x,c))≤cost(xi,F((→x−S,→yS),c)).

A mechanism is strongly group strategyproof for a class of cost functions if there is no coalition of agents misreporting their locations where at least one agent in benefits and the other agents in do not lose from the deviation. Formally, is strongly group strategyproof if for all cost functions and all location profiles , there do not exist a non-empty coalition and a location profile for , such that for all ,

 cost(xi,F(→x,c))≥cost(xi,F((→x−S,→yS),c)),

and there exists some agent with

 cost(xi,F(→x,c))>cost(xi,F((→x−S,→yS),c)).

## 3 The Equal-Cost Mechanism

In this section, we present and analyze the Equal Cost mechanism. At the conceptual level, Equal Cost, or , in short, works by equalizing the expected cost of all agents. Given an instance of -Facility Location on the line, works as follows:

Step 1

It computes an optimal covering of all agent locations with disjoint intervals that minimizes the interval length (wlog., we assume that ).

Step 2

It constructs a random variable such that all locations have the same the expected connection cost .

Step 3

For every interval , places a facility at , if is odd, or at , if is even.

We proceed to establish the main properties of , summarized by the following theorem. For the proof, we examine, in the following sections, each step of the mechanism separately.

###### Theorem 3.1

For the class of all concave cost functions, Equal Cost is group strategyproof and achieves an approximation ratio of for the objective of Max Cost, and an approximation ratio of for the objective of Social Cost. Moreover, for every instance (, with concave, and every agent , .

### 3.1 Step 1: Partitioning the Instance in Intervals

We can compute the minimum feasible interval length by checking all possible candidate values. The value of is equal to the distance for some agent locations . So, there are at most candidate values for . For each candidate value , we can check feasibility and compute a covering of all locations in with intervals of length as follows:

While there are uncovered agents, find the leftmost uncovered agent , and create a new interval .

The above algorithm computes the minimum number of intervals of length to cover . If this number is at most , we set . We can also speed up the algorithm by binary search over the space of candidate values.

We observe that the partitioning into intervals of length is closely related to the optimal maximum cost . In fact, an optimal solution can be obtained by placing a facility at the midpoint of each interval. Thus, the cost of the optimal solution is .

### 3.2 Step 2: Constructing the Random Variable

We next show that for any given cost function , we can construct a family of random variables such the expected cost of every point in is the same. For convenience, we denote this cost as . We note that , for all . In particular, for , we get .

We assume that the cost function is piecewise-linear with pieces of length and growth rates , where is the growth rate in the interval . For all , and , because is strictly increasing and concave. Our result applies to general concave functions either by discretizing appropriately, or by solving a continuous analog of the homogeneous linear system below through an integral equation. The technical details are related to the solution of integral equations and are beyond the scope of this work.

The support of the random variable is every point and , for integer . We note that if is an integer, we have only points in the support, instead of points in general. The crucial observation is that the derivative of the expected cost function in every interval between consecutive points in the support must be . So, to compute the probability assigned to each point in the support of , we write a set of linear equations and unknowns (the probability of each point in the support) requiring that the derivative of the expected cost function in each interval is . So, we get the homogeneous linear system . If is an integer, the matrix is:

Namely, the elements of the matrix are , if , and , if , for all and , where denotes the growth rate of the piecewise-linear cost function at the support point .

If is not an integer, the elements of the matrix are , if , and , if , for all and . Thus,

 Λ=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝λ0−λ0−λ0−λ1−λ1−λ2−λ2…−λ⌊ℓ⌋−1−λ⌊ℓ⌋−1−λ⌊ℓ⌋λ0λ0−λ0−λ0−λ1−λ1−λ2−λ2…−λ⌊ℓ⌋−1−λ⌊ℓ⌋−1⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮λ⌊ℓ⌋λ⌊ℓ⌋−1λ⌊ℓ⌋−1………λ1λ1λ0λ0−λ0⎞⎟ ⎟ ⎟ ⎟ ⎟⎠

We now show that in both cases there is a unique symmetric probability distribution that satisfies the system of equations. For this purpose, we use the two lemmas below. The first lemma is about a class of diagonally dominant matrices. It shows that we can bring any such matrix in a triangular form by performing Gaussian elimination, such that all diagonal elements are positive and all off-diagonal elements are less than or equal to .

###### Lemma 1

Let be a , matrix so that , for all , , for all , and , for all . Then, by performing elementary row operations (Gaussian elimination) on , we can get a row-echelon form where , for all , , for all , and , for all .

###### Proof

We use induction on . The base case, where , is already in the desired form. Assuming that the lemma holds for , we show that it holds for .

We have that , with and all elements of and non-positive. With a single step of Gaussian elimination, we get . To conclude the induction step, we show that the submatrix satisfies the properties of the lemma. Since all elements of are non-negative, we still have , for all . So, we need to show that , for all columns , which also implies that , for all . For any column , we have that:

 q∑i=1B′i,j=q∑i=1(Bi,j−viuj/a)=q∑i=1Bi,j−ujaq∑i=1vi>−uj−uja(−a)=0.

For the last inequality, we use that , and the hypothesis that , which implies that and that . ∎

The next lemma shows that for the special class of matrices arising in our case, there is a solution to the homogeneous linear system that defines a probability distribution.

###### Lemma 2

Let be a matrix defined as , where is a sequence of positive numbers such that , for all , and for all . Then, the system has a symmetric solution with , , and . Moreover, there is a unique symmetric solution that satisfies these conditions.

###### Proof

We let , for . Then, the matrix can be written as:

Taking the difference of every pair of ’s consecutive rows, we obtain the matrix

 A′=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−d12a0−d1…−dn−1−d2−d12a0…−dn−2⋮⋮⋮⋮⋮−dn−2−dn−3…−d1−d2−dn−1−dn−2…2a0−d1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠

To establish the lemma, we first use Lemma 1 and show that (i) the nullspace of contains a unique symmetric probability vector , and then show that (ii) the particular vector is also in the nullspace of .

As for claim (i), we first show that each coordinate of any vector in the nullspace of can be expressed as a non-negative linear combination of the coordinates and . Formally, we show that for any coordinate of any solution of , there exist , such that . To this end, we consider the matrix , which is obtained from by moving the first column of to the end. We observe that satisfies the conditions of Lemma 1, since , and thus . Hence, by applying Gaussian elimination to , we get a matrix in a row-echelon form with , for all , , for all , and , for all . Moreover, the nullspace of essentially consists of the solutions to the homogenous linear system . More precisely, any solution of corresponds to a solution of , where , , …, , and vice versa.

Due to the special form of , we can find all solutions of by assigning values to the free variables and and performing backwards substitution so that we uniquely determine the values of the variables . Furthermore, due to the special form of , this procedure results in expressing each variable as a non-negative linear combination of and . Specifically, we can calculate , for all , from the equation . Solving for , we get , since and , for all . Moreover, all coefficients are non-negative because , for all , and . By induction, if every , , is a non-negative linear combination of and , the same holds for . Therefore, any coordinate of any solution to can be expressed as a non-negative linear combination of the free variables and . Due to the aforementioned correspondence between the solutions of and the solutions of , we obtain that for any coordinate of any solution to , there exist , such that .

Hence, the nullspace of is spanned by the vectors and determined by setting the free variables and to and to , respectively. By the discussion above, all the coordinates of and are non-negative. To conclude the proof of claim (i), we observe that due to the symmetry of the homogeneous linear system , we have that , for all . Therefore, there is unique symmetric vector in the nullspace of with norm equal to , namely the vector .

We proceed to show claim (ii), namely that the unique symmetric probability vector in the nullspace of is also in the nullspace of . To this end, we define the matrix

 M=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝100…1−110…0⋮⋮⋮⋮⋮0…−1100…0−11⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠

We observe that the determinant of is equal to , and thus is non-singular. Therefore, the linear system is equivalent to the linear system . So, we let and be the first and the last row of , and further observe that is a matrix with its first row equal to and its remaining rows in one-to-one correspondence to the rows of . Since is the unique symmetric probability vector satisfying , we only need to show that , which follows immediately from the symmetry of . This completes the proof of claim (ii) and the proof of the lemma. ∎

For every , the homogeneous linear system satisfies the conditions of Lemma 2. Hence, there exists a unique symmetric probability distribution such that the expected cost is the same for every location . Next, we think of this unique symmetric solution as a function of , and establish a nice continuity property of it.

To this end, we fix an integer , and show that the random variable converges in probability to the random variable , as . We observe that the linear system determining is the same for all . So, we let be the probability assigned to each integer point , . By symmetry, the probability assigned to each point , , is also . The limit is a random variable distributed according to a probability distribution that assigns probability to each integer point , . Since the distribution is symmetric and achieves the same expected cost for all points , it is, by Lemma 2, the unique distribution with these properties. Therefore, we have that . By the same argument, we can show that the random variable converges in probability to the random variable , as .

By the continuity property above, the expected cost at each location is a continuous function of . Moreover, the discussion above implies that for all , . Using these properties, we now show that is an increasing function of .

###### Lemma 3

The expected cost is an increasing function of the interval length .

###### Proof

Since is continuous, we only need to show that is increasing in each interval , where is any integer. To this end, we let , and consider any . Then, we have that:

 C(ℓ)=E[c(X(ℓ))]=m∑i=0pmi(c(i)+c(ℓ−i))

where the inequality holds because and the cost function is increasing. ∎

### 3.3 Step 3: Establishing Group Strategyproofness

We next prove that the random facility placement, in Step 3 of Equal Cost, is group strategyproof. The correlation of the facility placement, in Step 3, ensures that if an agent is located at , his closest facility is always the one assigned to his closest interval. To justify this, let us consider any sample of the random variable . We recall that the facilities are placed at . Let us assume that . Then, the distance of to is , while the distance of to is . Hence, agent prefers the facility at interval if and only if , i.e., the right endpoint of interval is closer to than the left endpoint of interval .

To show that Equal Cost is group strategyproof, we consider a coalition of agents that deviate to improve their cost. Let the original interval length, with respect to the true agents’ locations, be , and let the new interval length, after the deviation, be . We now consider the two possible outcomes when the agents misreport their locations:

Case where . Let be any agent. If ’s true location is covered by some interval of the new covering, incurs an expected cost of . Otherwise, agent incurs an expected cost no less than , which is greater than .

Case where . We consider the distance of any agent to the nearest midpoint of an interval. The locations of the truthful agents in are covered by some interval of the new covering. Hence, their distance to the nearest midpoint of some interval is at most . On the other hand, if we consider the true locations of all agents and any feasible covering of them with intervals, there is some agent whose distance to the midpoint of the interval covering him is at least . Therefore, there is an agent whose distance to the nearest midpoint of some interval in the new covering (after the deviation) is at least . Hence, agent must be in the deviating coalition , and his true location must not be covered by the intervals of the new covering. In this case, Lemma 4 below implies that the expected cost of agent after the deviation, which is , is at least as large as . This implies that Equal Cost is group strategyproof.

###### Lemma 4

For all , , , with , it holds that

 IE[c(b−a+X(2a))]≥IE[c(b−a′+X(2a′))]

Moreover, the inequality is strict, if the function is strictly concave.

###### Proof

Let be any integer. We only need to show that the lemma holds for all , with . For all such , , , we have that:

 IE[c(b−a+X(2a))] = m∑i=0pmi(c(b−a+i)+c(b+a−i)) ≥ m∑i=0pmi(c(b−a′+i)+c(b+a′−i))=IE[c(b−a′+X(2a′))]

where the inequality holds because and is concave. In fact, the inequality is strict if is strictly concave. ∎

### 3.4 Approximation Ratio

In this section, we analyze the approximation ratio of Equal Cost.

###### Lemma 5

For any concave cost function , any locations profile , and any agent , it holds that .

###### Proof

We let be the minimum interval length in Step 1 of Equal Cost, and let . We recall that . Moreover, we have that:

 C(ℓ)=m∑i=0pmi(c(i)+c(ℓ−i))≤m∑i=02pmic(ℓ/2)=c(ℓ/2)

where the inequality follows from the concavity of the cost function . ∎

###### Lemma 6

For every concave cost function , Equal Cost has an approximation ratio of at most for the objective of Max Cost.

###### Proof

Let be any instance with a concave cost function , and let be the minimum interval length in Step 1 of Equal Cost. In , every agent has a facility at distance at most to . On the other hand, . Therefore, the approximation ratio is at most:

 c(ℓ)c(ℓ/2)=c(ℓ)+c(0)c(ℓ/2)≤2c(ℓ/2)c(ℓ/2)=2,

where we use that , by normalization, and the concavity of . ∎

###### Lemma 7

For every concave cost function , Equal Cost has an approximation ratio of at most for the objective of Social Cost.

###### Proof

For every locations profile , . Then,

 SC(→x,c)=∑i∈Ncost(xi,EC(→x,c))≤nMC∗(→x,c)≤nSC∗(→x,c),

where the inequality follows from Lemma 5. ∎

## 4 Applications

In this section, we consider three typical examples of concave cost functions, and derive closed form solutions for the corresponding random variables .

Linear Functions. The literature mostly focuses on linear cost functions , where the agents’ cost is proportional to their distance to the nearest facility. In this case, has a nice closed form: it is either with probability or with probability . Then, the expected connection cost of any location is:

 c(x)/2+c(ℓ−x)/2=λx/2+λ(ℓ−x)/2=2λℓ/2,

which does not depend on .

Two-Piece Piecewise Linear Functions. For some , let the cost function be:

 c(d)={λ1dfor d≤1λ2d+(λ1−λ2)for d>1

To achieve the same expected cost at all locations, we find , let , and compute the probability distribution of by solving the following linear system:

 ⎛⎜ ⎜ ⎜ ⎜⎝λ1−λ1−λ1−λ2−λ2−λ2−λ2…−λ2−λ2−λ2λ1λ1−λ1−λ1−λ2−λ2−λ2−λ2…−λ2−λ2⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮λ2λ2λ2………λ2λ2λ1λ1−λ1⎞⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝cpm0pmmpm1pmm−1⋮pmmpm0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=0

Taking the difference between every two consecutive rows, as in Lemma 2, we find that:

 pmi=λ1−λ22λ1(pmi−1+pmi+1)    for all integers i 0≤i≤m,

where we define , for all integers . Then, the solution of the recurrence is:

 pmi=ρm+1−i1+ρm+1−i22∑m+1j=1(ρj1+ρj2)
 where\ \ \ ρ1=λ1+√λ21−(