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