Facility location games with distinct desires

Discrete Applied Mathematics xxx (xxxx) xxx

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Facility location games with distinct desires✩ ∗

Lili Mei a , Minming Li b , , Deshi Ye a , Guochuan Zhang a a b

College of Computer Science, Zhejiang University, China Department of Computer Science, City University of Hong Kong, Hong Kong

article

info

Article history: Received 30 November 2017 Received in revised form 24 November 2018 Accepted 6 February 2019 Available online xxxx Keywords: Facility location game Strategyproofness Approximation ratio Happiness

a b s t r a c t In facility location games, one aims at designing a mechanism to decide the facility location based on the addresses reported by all agents. In the facility location game, each agent wants to minimize his/her distance from the facility, while in the obnoxious facility game, each agent prefers to be as far away from the facility as possible. In this paper we revisit the two games on a line network by finely defining another reasonable agent utility function in terms of their satisfaction degree with respect to the facility location. Namely, a happiness factor within [0, 1] is introduced to measure the difference between the best facility location for an agent and the one given by the mechanism. An agent wants to gain a largest possible happiness factor while the social satisfaction is to maximize the sum of the factors. For the facility location game, we first show that the median mechanism (Procaccia and Tennenholtz, 2009) is 32 -approximation. We then devise a 45 -approximate group strategyproof mechanism. Subsequently, we investigate the variant of maximizing the smallest happiness factor over all agents for this setting. For the obnoxious facility game, we show that the majority mechanism (Cheng et al., 2011) remains the best possible with approximation ratio of two. Finally, we devise a 43 -approximate randomized mechanism. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Facility location is one of the fundamental optimization problems, which allocates one or several facilities on a given network so that all clients in the network are served and the total cost is minimized. It is assumed that the full information of clients is known as public knowledge. In a game setting, however, the location of each client (we call an agent in the terminology of game theory) is private, which is not known before an algorithm is designed. The system manager is required to publicize an algorithm (mechanism) first. Then the agents report their locations (addresses) in the network, based on how the mechanism decides the facility locations. In the facility location game, each agent wants to minimize his/her distance from the facility, while in the obnoxious facility game, each agent prefers to be as far away from the facility as possible. It is of great interest if all agents are willing to tell the truth. The system manager thus aims at a strategy-proof mechanism maximizing the total utilities (or minimizing the total costs) of the agents. In the literature, the research of the facility location game and the obnoxious facility game has a rich history. However, to the best of our knowledge, in both of the facility location game and the obnoxious one, the cost or utility of each agent is simply a function of the distance from himself/herself to the facility location. However, for agents at different positions, the best utilities (or the worst costs) they can achieve are different, which makes them not equally happy even if the distances ✩ This work was supported by National Natural Science Foundation of China [Project NO. 11771365, 11671355, and 11531014]. ∗ Corresponding author. E-mail addresses: [email protected] (L. Mei), [email protected] (M. Li), [email protected] (D. Ye), [email protected] (G. Zhang). https://doi.org/10.1016/j.dam.2019.02.017 0166-218X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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are the same. For example, houses on a street usually have different prices. Those closer to the center of the street have easier access to nearby facilities and therefore will have a higher price. Thus, if a person bought a house in the center of the street, he would not be very happy if the important facility is built at an end of the street (since he already paid a lot at the time of purchase). On the other hand, if a person bought a house at an end of the street with a lower price, he would be happier if the facility is built in the center of the street, although the distances in these two cases are the same. To reflect the relative happiness of each agent compared to the best he/she can achieve, in this paper, we introduce the happiness factor of each agent (to be defined in the next section), which measures the agent’s degree of satisfaction for the facility location. Related works. The study of mechanism design addresses two categories of issues. One is characterizing the strategy-proof mechanisms and the other is designing strategy-proof mechanisms with good performance. For the facility location game, each agent wants to minimize the cost. The first result for this problem dates back to the 80s. Moulin [14] characterized all the anonymous, strategy-proof and efficient mechanisms for the single-peaked preference. Schummer and Vohra [16] studied the characterizations of all the strategy-proof mechanisms on other networks. Fotakis and Tzamos [9] characterized the strategy-proof mechanisms with bounded approximation ratio to minimize the total cost function for 2-facility location game on the line. Recently, Filos-Ratsikas et al. [7] considered the facility location game with double-peaked preference. Approximate mechanism design without money for the facility location game was first studied by Procaccia and Tennenholtz [15]. They proposed best possible strategy-proof mechanisms for the facility location game of minimizing the sum of all the agents’ costs (minSum) and the maximum cost (minMax). They also extended the facility location game to 2-facility location games and multiple locations per agent model. Alon et al. [1] extended the facility location game to other networks for minSum and minMax objectives. Lu et al. [12,13] improved some results of 2-facility location game and multiple locations per agent model for maxSum objective. Then Escoffier et al. [5] proposed (n − 1)-facility location game. Subsequently, Fotakis and Tzamos [10] studied k-facility location game with k ≥ 3. For the obnoxious facility game, where each agent wants to stay far away from the facility, Ibara and Nagamochi [11] characterized the strategy-proof mechanisms. For the approximation mechanism design direction, Cheng et al. [2] proposed and studied the obnoxious facility game for maximizing the sum of all the agents’ utilities (maxSum). Subsequently, Cheng et al. [3] extended the model to other networks. Recently, Zou and Li [20] and Feigenbaum et al. [6] considered the model where some agents want to stay close to the facility and the other agents want to stay far away from it. Zou et al. [20] also dealt with the problem where two facilities need to be located with a limitation on the relative distance and agents want to stay close to one of the facilities but be far away from the other facility. Serafino and Ventre [17,18] discussed the setting that each agent prefers one of the two facilities or both of them. In this model, the cost of the agent is the sum of the distances to the facilities she/he prefers and the social objective is to minimize the sum of all the agents’ costs or the maximum agent cost. Yuan et al. [19] studied the model that each agent has the optional preference for two heterogeneous facilities for the facility location game and the obnoxious facility game, respectively. The cost (or utility) of each agent is the minimum (or maximum) distance to the facilities he/she prefers. In [8], Fong et al. proposed a setting where each agent has a fractional preference to all the facilities. Our contributions. In this paper, we revisit the facility location game and the obnoxious facility game on a line network by finely defining another reasonable agent utility function in terms of their satisfaction degree with respect to the facility location. Namely, a happiness factor within [0, 1] is introduced to measure the difference between the best facility location for an agent and the one given by the mechanism. For the facility location game, we first show that the approximation ratio of the median mechanism is 32 for maximizing the sum of all the agents’ happiness factors, which is optimal for the minSum objective in [15]. We then explore a better group strategy-proof mechanism with approximation ratio 45 . This mechanism is executed on a closed interval setting. If the interval is open, however, this mechanism is not strategy-proof any more. And we show a lower bound of 1.086 for this problem. Moreover, if the number of agents is two, we establish a 1.07-approximate group strategy-proof mechanism and this mechanism is best possible. Finally, we briefly discuss the variant of maximizing the smallest happiness factor over all agents. For the obnoxious facility game, we show that the majority mechanism proposed by Cheng et al. [2] is 2-approximate, and this mechanism is best possible. Finally, we devise a strategy-proof randomized mechanism with approximation ratio of 34 . Organization of our paper. In the remainder of this paper, we define the models formally in Section 2. In Sections 3 and 4, we study the facility location game and the obnoxious facility game, respectively. Section 5 concludes this paper. 2. Problem formulation In the real scenarios, the facility locations are constrained by physical borders and agents are asked to specify their locations within these bounded domains. For instance, when a university plans to build a gymnasium, it makes sense to assume that the gymnasium will be within this university. Therefore, we first study the setting where the underlying network is a closed interval. Without loss of generality, we denote the interval as I = [0, 1]. There are n agents in the interval and let N = {1, 2, . . . , n} be the set of agents. Each agent i has a location xi ∈ I, which is private information. Let x = (x1 , x2 , . . . , xn ) ∈ I n denote the agents’ location profile. In our setting, given a collected location profile, a facility location on the interval is output. Let y ∈ I denote the facility location. Since our setting is an interval, the distance between agent Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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i and the facility location y, denoted by d(y, xi ), is |y − xi |. Each agent wants to maximize his/her satisfaction degree, called the happiness factor. Let dimax = max{d(0, xi ), d(1, xi )} denote the longer distance from agent i to two endpoints. For the facility location game where an agent wants to stay as close to the facility as possible, there are two extreme points. One is the location of the agent, while the other is the location farthest from the agent. The agent will be completely (100%) satisfied if the former is selected, and he/she will be least (0%) satisfied if the latter is chosen. Namely, we define his/her happiness factor on the two facility locations to be 1 and 0, respectively. Generally, the happiness factor function h(y, xi ) for agent i is defined below (recall that y is the facility location), h(y, xi ) = 1 −

d(y, xi ) dimax

.

(1)

Analogously, for the obnoxious facility game where each agent wants to stay as far from the facility as possible, the happiness factor function for agent i is given as follows, h(y, xi ) =

d(y, xi ) dimax

.

(2)

Note that the agent has a happiness factor of zero if the location is exactly where he/she is, and he/she has a happiness factor of one if the location is at the farthest point (the most preferable place). A (deterministic) mechanism f for our setting is a function f : I n → I where the input is the reported location profile x and the output is a facility location y on the interval I. In this ∑ paper, we mainly focus on the social satisfaction which is the sum of all the agents’ happiness factors, i.e., SH(y, x) = i∈N h(y, xi ). We aim to design mechanisms to elicit the true location profile (the reported location may be different from the true one) and maximize the social satisfaction as much as possible. If a mechanism f elicits the true location profile, we say that the mechanism is strategy-proof (or truthful), i.e., no agent can improve his/her happiness factor by misreporting. Formally, given any location profile x, for all x′i ∈ I, we have h(f (x), xi ) ≥ h(f (x′i , x−i ), xi ), where x−i = (x1 , . . . , xi−1 , xi+1 , . . . , xn ) is the location profile without agent i. Moreover, a mechanism f is group strategy-proof if no coalition of agents can improve their happiness factors by misreporting their locations simultaneously. Formally speaking, given any location profile x, for any non-empty subset S ⊆ N, there exists i ∈ S such that h(f (x), xi ) ≥ h(f (x′S , x−S ), xi ), for any x′S ∈ I |S | , where x−S is the location profile without agents in S. Let y∗ denote the optimal facility location. A mechanism f is ρ -approximation if SH(y∗ , x) ≤ ρ SH(f (x), x) for any location profile x. 3. Facility location game In this section, we first consider the setting where the objective function is maximizing the social satisfaction. Subsequently, we discuss the variant of maximizing the smallest happiness factor over all agents for this setting. 3.1. Strategy-proof mechanisms Median mechanism. It is worth noting that the happiness factor function is a single-peaked function. Hence the median mechanism in [15] is also group strategy-proof. In [15], the median mechanism gives an optimal facility location for minimizing the total costs of all agents (recall that the cost of an agent is the distance from himself/herself to the facility). However, in our setting the median is not always the optimal facility location. We first figure out the approximation ratio of the median mechanism (see below). The proof of the approximation ratio is inspired by two agent location profile where the optimal facility location is at one agent’s position while the mechanism outputs the other one. Then for a general location profile x = (x1 , . . . , xn ), we consider the approximation ratio for each pair of agents i and n − i + 1 with locations xi and xn−i+1 , i = 1, 2, . . . , ⌊ 2n ⌋. Mechanism 1 (Median Mechanism). Given a location profile x ∈ I n , without loss of generality, assume that x1 ≤ x2 ≤ · · · ≤ xn . Output f (x) = med(x), where med(x) = x⌈ n ⌉ . 2

Theorem 1. Mechanism 1 is a group strategy-proof mechanism with approximation ratio

3 2

for maximizing the social satisfaction.

Proof. Since the happiness factor function is single-peaked, Moulin [14] showed f (x) = med(x) is group strategy-proof. Hence we only need to show the approximation ratio of Mechanism 1 is 23 . Let y∗ be the optimal facility location. If med(x) = y∗ , Mechanism 1 achieves the optimal solution. We thus can assume that f (x) = med(x) ̸ = y∗ . Consider any pair of agents pi = (xi , xn−i+1 ), where i = 1, . . . , ⌊ 2n ⌋. To state conveniently, let SH(y, pi ) denote the sum of the happiness factors for agents i and n − i + 1 when a facility is located at y, where i = 1, . . . , ⌊ 2n ⌋. Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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If n is even, then we can see that any location profile x is equal to (p1 , . . . , p n ). Hence, for any location y ∈ I, we have 2

SH(y, x) = SH(y, p1 ) + · · · + SH(y, p n ), 2

which implies that SH(y∗ , x) SH(f (x), x)

{ SH(y∗ , pi ) } . i=1,..., 2 SH(f (x), pi )

≤ maxn

If n is odd, then x = (p1 , . . . , p⌊ n ⌋ , x⌈ n ⌉ ). Similarly, we have that for any location y ∈ I, SH(y, x) = SH(y, p1 ) + · · · + 2 2 SH(y, p⌊ n ⌋ ) + h(y, x⌈ n ⌉ ). Analogously, we get that 2

2

{ SH(y∗ , p ) ⏐ h(y∗ , x⌈ n ⌉ ) } i ⏐ 2 ≤ max , . ⏐ SH(f (x), x) SH(f (x), pi ) i=1,...,⌊ 2n ⌋ h(f (x), x⌈ n ⌉ ) 2 SH(y∗ , x)

h(y∗ ,x

)

We first show that h(f (x),x⌈n/2⌉ ) < 1. Note that if n is odd, the facility location returned by Mechanism 1 is med(x) = x⌈ n ⌉ , 2 ⌈n/2⌉ thus h(f (x), x⌈ n ⌉ ) = 1. Due to the assumption that f (x) ̸ = y∗ , we get that h(y∗ , x⌈ n ⌉ ) < 1. From the above, we prove the 2 2 result. If we can show that for all i = 1, . . . , ⌊ 2n ⌋, SH(y∗ , pi ) SH(f (x), pi )

3

≤

2

,

then the proof is complete. Let y∗i denote the optimal facility location when we only consider two agents in pair pi , where i = 1, . . . , ⌊ 2n ⌋. Note that SH(y∗ , pi ) ≤ SH(y∗i , pi ) and f (x) = med(x) ∈ [xi , xn−i+1 ]. Hence we get that for all i = 1, . . . , ⌊ 2n ⌋, SH(y∗ , pi ) SH(f (x), pi )

SH(y∗i , pi )

≤

min

y∈[xi ,xn−i+1 ]

{SH(y, pi )}

.

(3)

It is sufficient to show that for all i = 1, . . . , ⌊ 2n ⌋, the righthand side of inequality (3) is at most 32 . We use yˆ i to denote one position where we get minimum of SH(y, pi ) for all y ∈ [xi , xn−i+1 ], i.e., yˆ i = arg miny∈[xi ,xn−i+1 ] {SH(y, pi )}. Without loss of generality, we assume that xn−i+1 is closer to 12 than xi and xi ≤ 12 , otherwise, we construct a new pair (x¯ i , x¯ n−i+1 ) = (1 − xn−i+1 , 1 − xi ), and consider this new pair. According to the definition of happiness factor (equality (1)), we know that y∗i = xn−i+1 and yˆ i = xi . Combining with inequality (3), we have, for all i = 1, . . . , ⌊ 2n ⌋, SH(y∗ , pi ) SH(f (x), pi )

SH(xn−i+1 , pi )

≤

SH(xi , pi ) SH(xn−i+1 ,pi ) SH(xi ,pi )

Then we calculate If xn−i+1 ≤

1 , 2

.

(4)

according to xn−i+1 ≤

1 2

and xn−i+1 >

1 , 2

respectively (see Fig. 1 for an illustration).

then

SH(xn−i+1 , pi )

2−

=

SH(xi , pi )

2−

=

xn−i+1 −xi 1−xi xn−i+1 −xi 1−xn−i+1

(1 − xn−i+1 )(2 − xn−i+1 − xi ) (1 − xi )(2 − 3xn−i+1 + xi )

.

To state conveniently, we define a function g(v, w ) = (1−v )(2−3w+v ) , 0 ≤ v ≤ w ≤ g(xi , xn−i+1 ). We obtain that g(v, w ) is increasing on w , since (1−w )(2−w−v )

1 . 2

Then

SH(xn−i+1 ,pi ) SH(xi ,pi )

is exactly

∂ g(v, w) (w − v )(4 − v − 2w ) = > 0. ∂w (1 − v )(2 − 3w + v )2 Hence,

SH(xn−i+1 ,pi ) SH(xi ,pi )

= g(xi , xn−i+1 ) ≤ g(xi , 21 ) =

Therefore, if xn−i+1 ≤ SH(xn−i+1 , pi ) SH(xi , pi )

1 , 2

3 −xi 2

1+xi −2x2i

. Analogously, g(xi , 12 ) achieves the maximum value when xi = 0.

we have 1

3

2

2

≤ g(0, ) =

.

(5)

Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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Fig. 1. (a) xn−i+1 ≤

Finally, we consider the case that xn−i+1 > SH(xn−i+1 , pi )

2−

=

SH(xi , pi )

2−

=

1 . 2

1 2

and (b) xn−i+1 >

5

1 . 2

In this case,

xn−i+1 −xi 1−xi xn−i+1 −xi xn−i+1

xn−i+1 (2 − xi − xn−i+1 ) (1 − xi )(xn−i+1 + xi )

.

We use the same notations as in the case xn−i+1 ≤ decreasing on w due to

1 . 2

Then g(v, w ) =

w (2−v−w ) (1−v )(w+v )

, v ∈ [0, 12 ), w ∈ ( 12 , 1]. The function g is

∂ g(v, w) 2v − v 2 − 2vw − w 2 = ∂w (1 − v )(v + w )2 −(v − 12 )2 < ≤ 0. (1 − v )(v + w )2 The above inequality holds since w > SH(xn−i+1 , pi ) SH(xi , pi )

1

< g(xi , ) = 2

1 . 2

Therefore,

3 1 x 2 i 4 1 x2i x 2 i

−

−

−

1 2

.

Similarly, we know that g(xi , 12 ) is decreasing on xi . Hence, for xn−i+1 ≤ SH(xn−i+1 , pi ) SH(xi , pi )

1

3

2

2

< g(0, ) =

1 , 2

we have

.

(6)

It is worth noting that the approximation ratio of Mechanism 1 is tight with the following location profile where 2n agents are at 0 and 2n agents are at 12 . For this profile the optimal facility is at 12 and the optimal social satisfaction is 34 n. However, Mechanism 1 locates the facility at 0 and the social satisfaction is 21 n. The approximation ratio is 32 . Moreover, if the number 1 1 of agents is odd, consider the profile with n+ agents at 0 and n− agents at 12 . The approximation ratio of median mechanism 2 2 3 2 3 is of 2 − n+2 , which approaches 2 when the number of agents is infinity. □ Note that the profile of all the agents at 0 and the one with a half agents at 0 and the rest at 21 have the same median point. From the characterization in [14], we can intuitively focus on mechanisms with respect to the median position. Note that the latter is the worst case instance. If we want a better approximation ratio, then the mechanism needs to output another point rather than 0, but for the former one, if the mechanism outputs the same point as the former profile, then this point cannot be far away from 0. Hence, we design a mechanism like this: output y if med(x) ∈ [0, y]; output 1 − y if med(x) ∈ [1 − y, 1]; otherwise, output med(x). Then we get the following mechanism. Mechanism 2. Given a location profile x ∈ I n , without loss of generality, we assume that x1 ≤ x2 ≤ · · · ≤ xn . Output

⎧ ⎨ 1/5 f (x) = 4/5 ⎩

med(x)

if med(x) ∈ [0, 15 ] if med(x) ∈ [ 45 , 1] , otherwise

where med(x) = x⌈ n ⌉ . 2

We show that Mechanism 2 is group strategy-proof. Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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Lemma 1. Mechanism 2 is group strategy-proof. Proof. Let S ⊆ N be a coalition. We must demonstrate that the agents in S cannot all gain by misreporting, i.e., there exists an agent i ∈ S whose utility cannot strictly improve by simultaneously lying with agents in S. Let x′ = (x′S , x−S ). It is easy to see that if med(x) = med(x′ ), then the misreporting cannot happen. Hence, we only need to consider that med(x) ̸ = med(x′ ). Without loss of generality, we can assume med(x) < med(x′ ), which implies that f (x) ≤ f (x′ ). If f (x) = f (x′ ), then the misreporting also cannot happen. Now we need to focus on the case that f (x) < f (x′ ). Recall that in this case med(x) < med(x′ ), which implies that there must exist an agent i in the coalition S such that xi ≤ med(x) and x′i > med(x), otherwise med(x′ ) cannot be bigger than med(x). For agent i, we see that d(f (x), xi ) = f (x) − xi < f (x′ ) − xi = d(f (x′ ), xi ). It implies that h(f (x), xi ) > h(f (x′ ), xi ), that is to say, agent i strictly loses. □ Similar to the proof of Theorem 1, for a location profile x = (x1 , . . . , xn ), we also consider the approximation ratio of each pair of agents i and n − i + 1, i = 1, 2, . . . , ⌊ 2n ⌋. The difference is that the proof for each pair is more complicated for Mechanism 2. Lemma 2. The approximation ratio of Mechanism 2 is at most

5 4

for maximizing the social satisfaction.

Proof. We use the same notations as in the proof of Theorem 1. Recall that y∗ is the optimal facility location. For i = 1, . . . , ⌊ 2n ⌋, pi = (xi , xn−i+1 ) is the location profile of a pair of agents i and n − i + 1. Let SH(y, pi ) denote the sum of the happiness factors of location profile pi when a facility is located at y. Let f (x) denote the facility location output by Mechanism 2. SH(y∗ ,x) We already know that if n is odd then the approximation ratio SH(f (x),x) is at most max

{ SH(y∗ , p ) ⏐ h(y∗ , x⌈ n ⌉ ) } i ⏐ 2 , . ⏐ SH(f (x), pi ) i=1,...,⌊ 2n ⌋ h(f (x), x⌈ n ⌉ ) 2

If n is even, then SH(y∗ , x) SH(f (x), x)

≤ maxn { i=1,..., 2

SH(y∗ , pi ) SH(f (x), pi )

}.

Using the similar analysis to that in Theorem 1, we need to show that the above upper bounds of most 45 . We first show that h(y∗ , x⌈ n ⌉ ) 2

h(f (x), x⌈ n ⌉ )

≤

2

5 4

SH(y∗ ,x) SH(f (x),x)

,

are both at

(7)

where n is odd. If f (x) = med(x), it is easy to see that h(y∗ , x⌈n/2⌉ )/h(f (x), x⌈n/2⌉ ) ≤ 1. Next, we consider the case that f (x) = 51 or 45 . Without loss of generality, we assume that f (x) = 51 . Observe that in this case med(x) = x⌈n/2⌉ ≤ 51 . Hence, the happiness factor of agent ⌈n/2⌉ in Mechanism 2 denoted by h(f (x), x⌈n/2⌉ ) is 1 − h(y∗ , x⌈ n ⌉ ) 2

h(f (x), x⌈ n ⌉ )

=

2

≤ ≤

5 4 5 4 5 4

1/5−x⌈n/2⌉ 1−x⌈n/2⌉

=

5/4 . 1−x⌈n/2⌉

Therefore, the ratio

(1 − x⌈ n ⌉ )h(y∗ , x⌈ n ⌉ ) 2

2

(1 − x⌈ n ⌉ ) 2

.

The first inequality holds since the happiness factor is at most 1. The last inequality holds due to x⌈ n ⌉ ≥ 0. 2 Now, we need to prove that for any pair of agents pi , where i = 1, . . . , ⌊ 2n ⌋, SH(y∗ , pi ) SH(f (x), pi )

≤

5 4

.

(8)

Recall that y∗i denotes the optimal facility location for agents i and n − i + 1. Naturally, SH(y∗ , pi ) ≤ SH(y∗i , pi ). Hence, if we can show that for all i = 1, . . . , ⌊ 2n ⌋, SH(y∗i , pi ) SH(f (x), pi )

≤

5 4

,

(9)

then the proof is complete. Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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Fig. 2. (a) xn−i+1 ≤

1 ; 5

(b) xn−i+1 ∈ ( 15 , 12 ]; (c) xn−i+1 >

7

1 . 2

Similarly, we assume that xn−i+1 is closer to 21 than xi and xi ≤ 21 . Combining with the assumption, we can see that yi = xn−i+1 . We first deal with the case that xi ≥ 15 . By the assumption that xn−i+1 is closer to 21 , in this scenario, Mechanism 2 outputs med(x), i.e., f (x) = med(x). The analysis in the proof of Theorem 1 also holds. Therefore, we establish the following inequalities. ∗

SH(y∗i , pi ) SH(f (x), pi )

= ≤

SH(xn−i+1 , pi ) SH(med(x), pi ) SH(xn−i+1 , pi ) min

y∈[xi ,xn−i+1 ]

{SH(y, pi )}

=

SH(xn−i+1 , pi )

≤

SH( 12 , ( 15 , 12 ))

=

SH(xi , pi ) SH( 51 , ( 15 , 12 )) 65 56

<

5 4

. SH(x

,p )

i+1 i with regard to the locations of xn−i+1 dividing the locations Now we turn to the case that xi < 15 . We discuss SH(fn−(x) ,p i ) of xn−i+1 into three cases (see Fig. 2 for an illustration).

Case 1. xn−i+1 ≤

1 . 5

In this scenario, note that med(x) ∈ [xi , xn−i−1 ] ⊆ [0, 51 ]. Hence, f (x) =

SH(xn−i+1 , pi ) SH( 15 , pi )

2−

= 2−

xn−i+1 −xi 1−xi

1 −xi 5

1−xi

−

1 −xn−i+1 5

SH( 15 , pi )

2−

= ≤

and the ratio

.

1−xn−i+1

To simplify the righthand side of the above equation, let ω(z) = SH(xn−i+1 , pi )

1 5

1 −z 5

1−z

. Then,

xn−i+1 −xi 1−xi

2 − ω(xi ) − ω(xn−i+1 ) 2 2 − ω(xi ) − ω(xn−i+1 )

.

We observe that ω(z) is a decreasing function. Therefore, SH(xn−i+1 , pi ) SH( 15

, pi )

≤

1 1 − ω(0)

=

5 4

.

Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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Case 2. xn−i+1 ∈ ( 51 , 12 ]. Observe that in this case f (x) ∈ [ 15 , xn−i+1 ]. Hence, SH(xn−i+1 , pi ) SH(f (x), pi )

≤

SH(xn−i+1 , pi ) miny∈[ 1 ,x 5

SH(y, pi )

n−i+1 ]

.

Recall that according to the assumption xn−i+1 is the optimal facility location for agents in pi . By the definition of the happiness factor, the location 15 gets the minimum social satisfaction in [ 51 , xn−i+1 ]. Thus, SH(xn−i+1 , pi ) SH(f (x), pi )

≤

SH(xn−i+1 , pi ) SH( 15 , pi )

.

Then we find the upper bound of SH(xn−i+1 , pi ) SH(1/5, pi )

2−

= 2−

xn−i+1 −xi 1−xi

1/5−xi 1−xi

2+ Let ρ (v, w ) =

4 ( 1 5 1−xi

1+ 11−w −v 1 1 − 1−w ) 2+ 45 ( 1−v

xn−i+1 − 15 1−xn−i+1

1−xn−i+1 1−xi

1+

=

−

1 ) 1−xn−i+1

−

.

, v ∈ [0, 15 ), w ∈ ( 15 , 12 ]. Then

ρ (v, w) is increasing on w since ∂ρ (v, w) = ∂w

4 ( 1 5 (1−w )2

−

1 ) (1−v )2

SH(xn−i+1 ,pi ) SH( 15 ,pi )

is equal to ρ (xi , xn−i+1 ). Moreover, we can see that

w− 1

+ 2 (1−v)(15−w)

1 (2 + 45 ( 1−v −

1 ))2 1−w

≥ 0.

Hence, SH(xn−i+1 , pi ) SH( 15 Moreover,

, pi )

15−10xi 12−4xi

15 − 10xi

2

12 − 4xi

.

is decreasing on xi . Therefore, we get

SH(xn−i+1 , pi ) SH( 15

1

≤ ρ (xi , ) =

, pi )

1

5

2

4

≤ ρ (0, ) =

.

Case 3. xn−i+1 ≥ 21 . In this scenario, Mechanism 2 outputs f (x) ∈ [ 15 , min{ 54 , xn−i+1 }]. We use D to denote the interval [ 15 , min{ 45 , xn−i+1 }]. Let yˆ i denote a facility location where we can achieve miny∈D {SH(y, pi )}. Due to D ⊂ [xi , xn−i+1 ], y∗i = xn−i+1 and the definition of happiness factor, we know that yˆ i = 15 . From the above, we have SH(y∗i , pi ) SH(f (x), pi )

≤

SH(xn−i+1 , pi ) SH( 15 , pi )

.

Then we specialize SH(xn−i+1 , pi ) SH( 15 , pi )

2−

= 1−

xn−i+1 −xi 1−xi

1 − xi 5

+

1 5

1−xi xn−i+1 xn−i+1 −xi 2 1−xi

−

= 5·

4 1−xi

Similarly, let γ (v, w ) =

+

2− w−v 1−v 4 1 +w 1−v

1 xn−i+1

.

, v ∈ [0, 15 ), w ∈ ( 12 , 1]. We see that

∂γ (v, w) 2(1 − 2w )(1 + w − v ) + v (1 − v + 2w ) = 4 ∂w w 2 (1 − v )2 ( 1−v + w1 )2 ≤ 0, Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

L. Mei, M. Li, D. Ye et al. / Discrete Applied Mathematics xxx (xxxx) xxx

9

which implies that function γ is decreasing on w . Hence, we have SH(xn−i+1 , pi ) SH( 15 , pi )

1

< 5 · γ (xi , ) 2

= ≤

15 − 10xi 12 − 4xi 5 4

.

The last inequality holds since

15−10xi 12−4xi

is decreasing on xi .

Using the same profiles as in the proof of Theorem 1, we can see that the upper bound of 54 for Mechanism 2 is also tight. Recall that the profile is 2n agents at 0 and 2n agents at 12 if the number of agents is even. When the number of agents is odd, 1 agents at 0 and the rest agents at 21 . For the even one, the approximation ratio of Mechanism 2 is just 54 . the profile is n+ 2 −1 For the odd one, the approximation ratio is 45 · 3n , which approaches 45 if n tends to infinity. □ 3n+1 By Lemmas 1 and 2, we have the following theorem. Theorem 2. Mechanism 2 is a group strategy-proof mechanism with approximation ratio

5 4

for maximizing the social satisfaction.

3.2. Lower bounds We initially consider the lower bound from profiles with two agents .

√

Theorem 3. No strategy-proof deterministic mechanisms have an approximation ratio less than 8 − 4 3 ≈ 1.07 for maximizing the social satisfaction. Proof. Let f be a strategy-proof deterministic mechanism. Consider a location profile x with two agents, where x1 = 0 and x2 = 1. Since agent 1 and agent 2 are symmetric of 12 , without loss of generality, assume that y = f (x) ∈ [ 21 , 1]. We then consider a new location profile x′ = (x′1 = α, x2 = 1), where 0 < α < 12 . The value of α will be specified later. By the strategyproofness, we get that f (x′ ) ∈ / [0, y), otherwise the agent at x1 in profile x can misreport to α and benefits. Similarly, we have that f (x′ ) ∈ / (y, 1], otherwise the agent at α in profile x′ can benefit by misreporting to 0. Hence, f (x′ ) = y. Denote y∗′ as the optimal facility location for location profile x′ . Since 0 < α < 12 , we can easily get that y∗′ = α and the optimal social satisfaction SH(y∗′ , x′ ) = 1 + α . The social satisfaction of the facility location output by mechanism √ f, y−α 1−α 2 1−α 2 SH(f (x′ ), x′ ), is 1 − 1−α + y. The approximation ratio is 1y+α ≥ . It is easy to see when α = 2 − 3 the = α −α 1−α y 1− 1− 1−α +y

2

√

righthand side of the inequality gets the maximum value and the approximation ratio is at least 8 − 4 3. □ Moreover, for two agents setting, we devise the following simple mechanism and show the mechanism is exactly √ (8 − 4 3)-approximate, which implies that we need to employ profiles with more than two agents if we intend to improve the lower bound. Mechanism 3. Given a location profile x ∈ I 2 , without loss of generality, assume that x1 ≤ x2 . Output f (x) =

⎧ ⎨ x2

x1

⎩

1 2

if x1 ≤ x2 ≤ 12 if 12 ≤ x1 ≤ x2 . otherwise

√ Theorem 4. Mechanism 3 is a group strategy-proof mechanism with approximation ratio 8 − 4 3 for maximizing the social satisfaction. Proof. It is obvious that the mechanism is group strategy-proof. We then focus on the approximation ratio. If x1 ≤ x2 ≤ 21 or 21 ≤ x1 ≤ x2 , Mechanism 3 gets the optimal facility location. Now let x1 < 12 < x2 . Without loss of generality, assume that 1 − x1 ≤ x2 , which implies that the optimal facility is at x1 . The approximation ratio SH(x1 , x) SH( 12

, x)

= 1+

x2 − 1 + x1 x2 + 1 − x1

= 1 + (1 − ≤ 1+

(1 − 2x1 )

2(1 − x1 )

x2 + 1 − x1 x1 (1 − 2x1 )

)(1 − 2x1 )

2 − x1

√

≤ 8−4 3 Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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The first inequality holds since x2 ≤ 1. The last one holds since the function g(z) =

√

2−

√

3 and g(2 −

√

z(1−2z) 2−z

achieves the maximum value at

3) = 7 − 4 3. □

Now, we explore better lower bounds with more agents’ profiles and establish the following theorem. Theorem√ 5. If the number of the agents is large, no strategy-proof deterministic mechanisms have an approximation ratio less than 25 − 2 ≈ 1.086 for maximizing the social satisfaction. Proof. We use f to denote a strategy-proof deterministic mechanism. Similar to the proof of Theorem 3, we design a profile αn

(1−α )n

      x with two locations 0 and 1. However, in this proof the profile is not symmetric. Let x = (0, . . . , 0, 1, . . . , 1) denote the profile, where α n agents are at location 0, the remaining agents are at 1 and 0 < α < 12 . The value of α will be calculated later. Due to 0 < α < 21 , the optimal facility location of profile x is at 1. Let y = f (x) denote the facility location output by mechanism f . Then the approximation ratio of mechanism f for profile x is 1−α

ρx (y) =

α (1 − y) + (1 − α )y

=

1−α

α + (1 − 2α )y

.

Note √that ρx (y) is decreasing with respect to y. We hope that ρx ≥

α ∈ (0,

5 2

√ −

2 if y ≤

1 . 2

This condition implies that

− ]. Hence, we can assume that y = f (x) > otherwise mechanism f already gets the lower bound. αn    (1−α)n 1 1    Then we establish another profile x′ = ( , . . . , , 1, . . . , 1). We claim that f (x′ ) = f (x) = y. We can view profile x′ as 2 2

1 4

1 , 2

2 2 the agents at 0 in profile x deviate to 12 one by one. Due to the strategyproofness, in each step, mechanism f returns facility y, which immediately implies that f (x′ ) = y. We expect to design a profile where the approximation ratio of mechanism f is strictly increasing on y. Hence, the optimal facility is at 21 for profile x′ , which is to say, α > 13 . Then the approximation ratio of mechanism f for profile x′ is

ρx′ (y) =

1+α 4α (1 − y) + 2(1 − α )y

=

1+α 4α + 2(1 − 3α )y

It is worth noting that ρx′ (y) is increasing on y. If y =

ρ≥

8α − 15α + 8α − 1 3

2

2α 3 − 4α 2 + 2α

.

5α 2 −3α , 8α 2 −7α+1

then ρx (y) = ρx′ (y). Hence, the lower bound

.

(10)

√

√

The righthand side of the above inequality reaches the maximum value when α = 2 − 1, and the lower bound is 52 − 2. It is worth noting that α is irrational. If the number of agents n is large enough, then the lower bound can infinitely approach √ 5 − 2. □ 2 Remark. It is interesting to narrow the gap between the lower bound 1.086 and the upper bound

5 4

for the general case.

3.3. Maximizing the minimum happiness factor In addition to maximizing the sum of happiness factors, it is also interesting to maximize the minimum happiness factor concerning about the fairness. We consider the setting where all the agents are within a closed interval I = [0, 1]. It is easy to see that the naive middle point mechanism (always picks the middle point 12 ) is a 2-approximation group strategy-proof mechanism, since for any agent i ∈ N, d( 21 , xi ) ≤ 12 dimax . On the other hand, it is not hard to show that no strategy-proof deterministic mechanism can achieve an approximation ratio better than 34 . A brief proof is stated below. Let f be a strategy-proof deterministic mechanism that can achieve approximation ratio better than 43 . Consider the location profile x with two agents where x1 = 0 and x2 = 1. The optimal solution will put the facility at 21 , achieving minimum happiness 12 . Therefore, we have 38 < f (x) < 58 in order to achieve the approximation ratio better than 43 . Without loss of generality, we assume that f (x) ≥ 12 . Then we consider profile x′ with x1 = 0 and x2 = f (x) (moving agent 2 to the left f (x) to the position of the facility). The optimal solution for x′ is f (x)+1 achieving the minimum utility f (x)1+1 . In order to achieve 3 the approximation ratio better than 34 , we have 4(f (x) < 1 − f (x′ ) which gives f (x′ ) < 12 since f (x) ≥ +1) misreport his true location f (x) to be 1 to improve his happiness, which gives a contradiction.

1 . 2

Then agent 2 can

4. Obnoxious facility game In this setting each agent wants to be as far away from the facility as possible. Hence the mechanism proposed in [2], which we call majority mechanism, is still strategy-proof. We present majority mechanism below, Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

L. Mei, M. Li, D. Ye et al. / Discrete Applied Mathematics xxx (xxxx) xxx

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Mechanism 4 (Majority Mechanism). Given a location profile x ∈ I n , let L denote the set of agents in [0, 12 ] and R be the set of agents in ( 21 , 1], respectively. Output f (x) = 0 if |L| ≤ |R| and otherwise output f (x) = 1. Recall that in the obnoxious facility game, the happiness factor for agent i is h(y, xi ) =

d(y, xi ) dimax

,

(11)

where xi is the location of agent i and y is the facility location determined by Mechanism 4. It is easy to see that every agent in L gets a happiness factor of one if y = 1, and every agent in R will be happiest if y = 0. Thus at least half of the agents will achieve their best possible choices. The approximation ratio for Mechanism 4 is two. Theorem 6. Mechanism 4 is a strategy-proof mechanism with approximation ratio 2 for maximizing the social satisfaction. Remark. Mechanism 4 can be extended to the tree network. Let T denote the underlying network. We first find a diameter da,b , which can be found in polynomial time. Here a and b denote two endpoints of da,b . Let cen(a, b) denote the midpoint of da,b , i.e., d(a, cen(a, b)) = d(b, cen(a, b)). We divide T into two subtrees by cutting cen(a, b). Let Ta (resp. Tb ) denote the subtree containing a (resp. b). If there are agents in the point cen(a, b), then let cen(a, b) belong to Ta . Let na (resp. nb ) denote the number of agents in Ta (resp. Tb ). If na ≥ nb , then the mechanism outputs b; otherwise the mechanism outputs a. It is easy to see that this mechanism is strategy-proof and the approximation ratio is also two. Now we consider the lower bound of the obnoxious facility game. We show the lower bound using the profile with one agent at 14 and the other agent at 43 . We can construct another profile whose approximation ratio is at least the lower bound according to the facility location output by any strategy-proof mechanism for profile ( 14 , 34 ). Theorem 7. Let N = {1, 2, . . . , n}, where n ≥ 2, and let ϵ > 0. Any strategy-proof deterministic mechanism has an approximation ratio at least 2 − ϵ for maximizing the social satisfaction. In order to prove the theorem, we need the following lemma. Lemma 3. Given a location profile x = (a, b) where a ≤ 21 , b ≥ 12 and the distance between a and b is at most 21 , let y denote the facility location. If y ∈ [a, b], the approximation ratio is at least 2 for maximizing the social satisfaction. Proof. For the profile x = (a, b), we first consider the optimal solution. Let y∗ denote the optimal facility location and SH(y∗ , x) be the optimal social satisfaction. Since a ≤ 12 and b ≥ 12 , without loss of generality, we can assume that a is closer to 12 than b. According to the definition of the happiness factor for the obnoxious facility game, we get that y∗ = 0 and a SH(y∗ , x) = 1− + 1. a Then we consider SH(y, x) for y ∈ [a, b]. Similarly by the assumption and the definition of the happiness factor, we a . conclude that SH(y, x) achieves the maximum social satisfaction when y = b, i.e., maxy∈[a,b] SH(y, x) = 1b− −a Hence, the approximation ratio ρ = and b is at most 12 , i.e., b − a ≤

1 . 2

SH(y∗ ,x) SH(y,x)

≥

a +1 1 −a b−a 1−a

=

1 b−a

≥ 2. The last inequality holds since the distance between a

□

Proof of Theorem 7. Let f be any strategy-proof deterministic mechanism. We consider a location profile x with two agents x1 = 41 and x2 = 34 . The optimal facility location of profile x is at either 0 or 1 and the optimal social satisfaction is 43 . If f (x) ∈ [ 14 , 34 ], then the social satisfaction of mechanism f , SH(f (x), x), is equal to 32 , which implies that the approximation ratio of mechanism f is already 2. Hence, we only need to consider that f (x) ∈ / [ 41 , 43 ]. Since the two agents are symmetric of 1 3 , without loss of generality, we can assume that f (x) ∈ ( , 1 ] . 2 4 We first discuss the case that f (x) ∈ ( 34 , 1). We find a positive integer k ≥ 2 such that f (x) ∈ (1 − 21k , 1 − 2k1+1 ]. Then for another location profile x′ = (x′1 = 12 − 2k1+1 , x2 = 43 ), we claim that f (x′ ) = f (x). By the strategyproofness, it is easy to see that f (x′ ) ≤ f (x), otherwise the agent at 14 in profile x can benefit by misreporting to 12 − 2k1+1 ; Analogously, we can see that f (x′ ) ∈ / [0, f (x)), otherwise the agent at x′1 in profile x′ can misreport to 14 and benefits. Next, we consider the following location profile x′′ = (x′1 = 12 − 2k1+1 , x′′2 = f (x)). By the strategyproofness of the agent at 43 in profile x′ , we say that d(f (x′′ ),

3 4

) = |f (x′′ ) −

3 4

3

| ≤ f (x) − , 4

that is to say, f (x′′ ) ∈ [ 32 − f (x), f (x)] ⊂ [ 12 − 2k1+1 , f (x)]. Note that 1 − 21k < f (x) ≤ 1 − 2k1+1 . The distance between f (x) and 21 − 2k1+1 , d( 12 − 2k1+1 , f (x)) ≤ 12 . By Lemma 3 with location profile x′′ , we can obtain that the approximation ratio of mechanism f is at least 2. Finally, we move to the case that f (x) = 1. For a given ϵ > 0, we establish a new profile xˆ = (xˆ 1 = 21 − 4−ϵ2ϵ , x2 = 34 ). By strategyproofness, we know that f (xˆ ) = 1. Then we consider one more profile xˆ ′ = (xˆ 1 = 21 − 4−ϵ2ϵ , xˆ ′2 = 1). By the Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

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strategyproofness of the agent at 34 , we can get that f (xˆ ′ ) ∈ [ 12 , 1], which means the social satisfaction of mechanism f is at most 1. For the profile xˆ ′ , the optimal facility location is at 0 and the optimal social satisfaction is 2 − ϵ . Hence, the approximation ratio of mechanism f is at least 2 − ϵ , where ϵ > 0. □ Randomized mechanisms For the obnoxious facility game, by Theorem 7, we can see that majority mechanism gives the best possible deterministic mechanism. Now, we turn to randomized mechanisms, which output a probability distribution P on the interval I = [0, 1]. Then the happiness factor of each agent i is h(P , xi ) =

Ey∼P d(y, xi ) dimax

.

∑n

And the social satisfaction is SH(P , x) = i=1 h(P , xi ). We establish the following randomized mechanism. Mechanism 5. Given a location profile x ∈ I n , let L denote the set of agents in [0, 12 ] and R be the set of agents in ( 12 , 1], respectively. |R| |L| Output 0 with probability n and 1 with probability n . Before showing the approximation ratio of Mechanism 5, we first discuss the optimal facility location of this setting. ∑n Recall that the happiness factor of each agent i is h(y, xi ) = d(y, xi )/dimax and the social satisfaction is SH(y, x) = i=1 h(y, xi ). For each agent i, we can view 1/dimax as the weight of agent i. Then for the optimization perspective, this setting becomes 1-maxian problem [4]. From [4], we know that there always exists an optimal facility location which is one of the endpoints. Theorem 8. Mechanism 5 is a strategy-proof mechanism with approximation ratio

4 3

for maximizing the social satisfaction.

Proof. Since agents in L prefer 1 to 0, and agents in R prefer 0 to 1. If an agent in L misreports, it can only improve the probability of location 0 (or the probability remains the same), which cannot increase the happiness factor of the agent. For the agents in R, the analysis is similar. Thus, no agent has incentive to misreport. Next, we show the approximation ratio of Mechanism 5 is 43 . Without loss of generality, we can assume that the optimal facility location is at 0. Thus , let OPT (x) = SH(0, x) be the optimal social satisfaction and P be the probability distribution output by Mechanism 5. The social satisfaction of mechanism 5 is SH(P , x) =

= ≥

|R| n |R| n |R| n

SH(0, x) + OPT (x) + OPT (x) +

|L| n

SH(1, x)

|L|

SH(1, x) n |L|2 n

The last inequality holds since if the facility is located at 1 then all the agents at [0, 12 ] get the happiness factor of 1, i.e., SH(1, x) ≥ |L|. Note that OPT (x) ≤ n. Thus,

( | L|

SH(P , x) ≥ (

)2 +

|R| )

OPT (x) n n ( |L| ) |L| = ( )2 − + 1 OPT (x) n n ( | L| 1 3) = ( − )2 + OPT (x) n 2 4 3 ≥ OPT (x), 4

which completes our proof. □ 5. Concluding remarks In this paper we revisit the two games on a line network by defining another reasonable agent cost (utility) function in terms of their satisfaction degree with respect to the facility location. For the facility location game, we first show the median mechanism proposed in [15] is a 32 -approximation group strategyproof mechanism. Then we explore a better group strategy-proof mechanism with approximation ratio 54 . Finally, we obtain √ the lower bound 25 − 2 for this problem. Moreover, we establish a best possible group strategy-proof mechanism for two agents. It is interesting to improve the lower bound of this problem or establish a strategy-proof mechanism with approximation ratio better than 54 . Subsequently, we briefly study the variant of maximizing the smallest happiness factor Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.

L. Mei, M. Li, D. Ye et al. / Discrete Applied Mathematics xxx (xxxx) xxx

13

over all the agents. Note that there is a big gap between the upper bound 2 and the lower bound 4/3. It is worth figuring out a tighter bound. For the obnoxious facility game, it is obvious that the majority mechanism proposed by Cheng et al. [2] is strategy-proof. We show that the majority mechanism is a 2-approximate mechanism. We also prove a matching lower bound. Finally, we devise a randomized mechanism with approximation ratio of 43 . For the future work, it is interesting to consider randomized mechanisms which output probability distributions of the facility location over I. Moreover, it is also interesting to extend our model to other networks, e.g., trees, circles or general metric spaces. Finally, it is also a possible direction to study the k-facility location problem, where k ≥ 2. References [1] Naga Alon, Michal Feldman, Ariel D. Procaccia, Moshe Tennenholtz, Strategyproof approximation of the minimax on networks, Mathematics of Operations Research 35 (3) (2010) 513–526. [2] Yukun Cheng, Wei Yu, Guochuan Zhang, Mechanisms for obnoxious facility game on a path, in: Proceedings of the 5th Annual International Conference on Combinatorial Optimization and Applications (COCOA), 2011, pp. 262–271. [3] Yukun Cheng, Wei Yu, Guochuan Zhang, Strategy-proof approximation mechanisms for an obnoxious facility game on networks, Theoretical Computer Science 497 (2013) 154–163. [4] Richard L. Church, Robert S. Garfinkel, Locating an obnoxious facility on a network, Transportation Science 12 (2) (1978) 107–118. [5] Bruno Escoffier, Laurent Gourvès, Nguyen Kim Thang, Fanny Pascual, Olivier Spanjaard, Strategy-proof mechanisms for facility location games with many facilities, in: Proceedings of the 2nd International Conference on Algorithmic DecisionTheory (ADT), 2011, pp. 67–81. [6] Itai Feigenbaum, Jay Sethuraman, Strategyproof mechanisms for one-dimensional hybrid and obnoxious facility location models, in: Workshops at the 29th AAAI Conference on Artificial Intelligence, 2015. [7] Aris Filos-Ratsikas, Minming Li, Jie Zhang, Qiang Zhang, Facility location with double-peaked preferences, Autonomous Agents and Multi-Agent Systems 31 (6) (2017) 1209–1235. [8] Ken C.K. Fong, Minming Li, Pinyan Lu, Taiki Todo, Makoto Yokoo, Facility location games with fractional preferences, in: Proceedings of the 32th AAAI Conference on Artificial Intelligence (AAAI), 2018, pp. 1039–1046. [9] Dimitris Fotakis, Christos Tzamos, On the power of deterministic mechanisms for facility location games, ACM Transactions on Economics and Computation 2 (4) (2014) 15:1–15:37. [10] Dimitris Fotakis, Christos Tzamos, Strategyproof facility location for concave cost functions, Algorithmica 76 (1) (2016) 143–167. [11] Ken Ibara, Hiroshi Nagamochi, Characterizing mechanisms in obnoxious facility game, in: Proceedings of the 6th Annual International Conference on Combinatorial Optimization and Applications (COCOA), 2012, pp. 301–311. [12] Pinyan Lu, Xiaorui Sun, Yajun Wang, Zeyuan Allen Zhu, Asymptotically optimal strategy-proof mechanisms for two-facility games, in: Proceedings of the 11th ACM Conference on Electronic Commerce (EC), 2010, pp. 315–324. [13] Pinyan Lu, Yajun Wang, Yuan Zhou, Tighter bounds for facility games, in: Proceedings of the 5th International Workshop on Internet and Network Economics (WINE), 2009, pp. 137–148. [14] Hervé Moulin, On strategy-proofness and single peakedness, Public Choice 35 (4) (1980) 437–455. [15] Ariel D. Procaccia, Moshe Tennenholtz, Approximate mechanism design without money, in: Proceedings of the 10th ACM Conference on Electronic Eommerce (EC), 2009, pp. 177–186. [16] James Schummer, Rakesh V. Vohra, Strategy-proof location on a network, Journal of Economic Theory 104 (2) (2002) 405–428. [17] Paolo Serafino, Carmine Ventre, Heterogeneous facility location without money on the line, in: Proceedings of the 21st European Conference on Artificial Intelligence (ECAI), 2014, pp. 807–812. [18] Paolo Serafino, Carmine Ventre, Truthful mechanisms without money for non-utilitarian heterogeneous facility location, in: Proceedings of the 29th AAAI Conference on Artificial Intelligence (AAAI), 2015, pp. 1029–1035. [19] Hongning Yuan, Kai Wang, Ken C.K. Fong, Yong Zhang, Minming Li, Facility location games with optional preference, in: Proceedings of the 23nd European Conference on Artificial Intelligence (ECAI), 2016, pp. 1520–1527. [20] Shaokun Zou, Minming Li, Facility location games with dual preference, in: Proceedings of the 14th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), 2015, pp. 615–623.

Please cite this article as: L. Mei, M. Li, D. Ye et al., Facility location games with distinct desires, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.017.