4.2 Sum Cost 5 CONSLUSION and DISCUSSION

1526 H. Yuan et al. / Facility Location Games with Optional Preference 1 ∪ 1 ∪ is located between them, which is the distance between them. Com- and 2 max(X1 X1,2) < 2 max(X1 X1,2). Therefore y1 = 1 1 ∪ ∪ bining every pair of agents with preference {F1} from out most to 2 max(X1 X1,2) < 2 max(X1 X1,2)=y1, also implying an increase in his cost. Therefore no agent has the incentive to lie. the center, we can see that the sum cost is minimized when {F1} is { } Case 3. p1 = {F2},pn = {F1}. It is not hard to see that we have located at the median location of all agents with preference F1 , the same result as Case 2 by symmetry. which is exactly the output of Mechanism 4. The analysis for agents Therefore Mechanism 3 is strategy-proof. with preference {F2} is similar. Therefore, Mechanism 4 is optimal for the fixed preference model. Theorem 14 Mechanism 3 is optimal. Now we present the proof for the approximation ratio of Mecha- nism 4. Proof. We will analyze by the same cases in the proof for Theorem 13. For Case 1, the maximum cost comes from agent 1 or agent n and Theorem 17 Mechanism 4 is 2-approximation. we have the maximum cost mc = L/2. We can see that any other Proof. For our optional preference (Max) model, we have the cost output would have the new maximum cost mc >L/2=mc.For as: Case 2, the maximum cost mc comes from agent 1 and the agent at max(X1 ∪ X1,2), or agent n and the agent at min(X2 ∪ X1,2).We scMax = d(i, Fk)+ max(d(i, F1),d(i, F2)) ∈{ } { } { } will focus on the former one as the analysis for the latter one would k 1,2 pi= Fk pi= F1,F2 be similar. Assume there is a better output denoted as y1 and y2.If while for the fixed preference model, we have the cost as y1 >y1, we can see the cost of agent 1 increases, which means that the maximum cost increases. If y1 <y1, indeed the cost of agent 1 scSum = d(i, Fk)+ (d(i, F1)+d(i, F2)) k∈{1,2} p ={F } p ={F ,F } decreases. However for the agent at max(X1 ∪X1,2) we would have i k i 1 2 ∪ − his cost mc be the maximum cost and mc = max(X1 X1,2) y1 > Since scMax(s) ≥ { } min(d(i, F1),d(i, F2)) we have: 1 ∪ pi= F1,F2 2 max(X1 X1,2)=mc, which means that the maximum cost also increases. Therefore no such output exists and Mechanism 3 is 2scMax ≥ scSum optimal. Based on this equation, given a profile, let s be the solution returned ∗ by Mechanism 4, and s be the optimal solution, we now show that ∗ 4.2 Sum cost scMax(s) ≤ 2scMax(s ). ∗ For minimizing sum cost, we will present a strategy-proof mecha- By Lemma 16 , we have scSum(s) ≤ scSum(s ). Therefore nism with approximation ratio of 2. ∗ ∗ scMax(s) ≤ scSum(s) ≤ scSum(s ) ≤ 2scMax(s ) Mechanism 4 Given a profile, locate F1 at the median location of Eventually, the theorem is proved. ∪ ∪ X1 X1,2 and F2 at the median location of X2 X1,2. Theorem 15 Mechanism 4 is strategy-proof. 5 CONSLUSION AND DISCUSSION Proof. Let agent i be the lying agent. If he has single preference, We studied the two heterogeneous facility location game with op- from Mechanism 4 we can see that he cannot change the output for tional preference and we mainly focused on deterministic mecha- { } his preferred facility by lying to have preference F1,F2 , and can nisms. This is a new model which covers more real life scenarios. A only possibly push it away by lying to have preference for the other table summarizing our results is listed below. facility since the mechanism is based on median location. If he has optional preference, by lying to have preference {F1}, he can only Table 1. A summary of our results. possibly push F2 away from him and cannot change the location of F1, which does not reduce his cost. Lying to have preference F2 will Variant Objective Upper Bound Lower Bound have similar effect. Therefore, no agent has the incentive to lie and maximum cost 2 4/3 Min Mechanism 4 is strategy-proof. sum cost n/2+1 2 maximum cost 1 (optimal) To prove the approximation ratio of Mechanism 4, let us first Max investigate a different model, referred to as the fixed preference sum cost 2 model. In this model, the cost of agent i is defined as costi = ∈ d(i, Fk). Without considering the truthfulness, we can also Fk pi Besides the above results, we also found that the approximation apply Mechanism 4 to this model. And we will have the following ratio can be better if randomized mechanisms are allowed. For ex- lemma. ample, the ratio of the Min variant for maximum cost can be low- Lemma 16 Mechanism 4 is optimal for the fixed preference model. ered to 3/2 and is likely to be a constant for sum cost. On the other hand, in our setting the two facilities can be located at any point on Proof. Given that the cost of agents with preference {F1,F2} de- the continuous line and can be located together, which is well jus- pends on the sum of distances from both facilities in fixed preference tified. However, it is also an interesting direction to study the case model, we can split them into an agent with preference {F1} and when facilities cannot be put on the same point. Some of the mecha- an agent with preference {F2} located at the same place and we will nisms proposed in our paper can be applied to that discrete case. For have a new profile containing only agents with preference for a single example, for the Min variant minimizing sum cost, the mechanism facility. It is not hard to see that the new profile is equivalent to the we proposed will never locate the two facilities together unless all original profile for analyzing the cost. Meanwhile, for any two agents agents are located together, and this mechanism could potentially be with preference {F1}, the sum of their cost is minimized when {F1} extended to the k-facility model. H. Yuan et al. / Facility Location Games with Optional Preference 1527 Acknowledgment This research was partially supported by NSFC 61433012 and a grant from the Research Grants Council of the Hong Kong Special Admin- istrative Region, China [Project No. CityU 117913]. REFERENCES [1] Noga Alon, Michal Feldman, Ariel D Procaccia, and Moshe Ten- nenholtz, ‘Strategyproof approximation of the minimax on networks’, Mathematics of Operations Research, 35(3), 513–526, (2010). [2] Yukun Cheng, Wei Yu, and Guochuan Zhang, ‘Mechanisms for obnoxious facility game on a path’, in Combinatorial Optimization and Ap- plications, 262–271, Springer, (2011). 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