Nearly Complete Characterization of 2-Agent Deterministic Strategyproof Mechanisms for Single Facility Location in L_p Space
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We consider the problem of locating a single facility for 2 agents in L_p space (1<p<∞) and give a nearly complete characterization of such deterministic strategyproof mechanisms. We use the distance between an agent and the facility in L_p space to denote the cost of the agent. A mechanism is strategyproof iff no agent can reduce her cost from misreporting her private location. We show that in L_p space (1<p<∞) with 2 agents, any location output of a deterministic, unanimous, translation-invariant strategyproof mechanism must satisfy a set of equations and mechanisms are continuous, scalable. In one-dimensional space, the output must be one agent's location, which is easy to prove in any n agents. However, in m-dimensional space (m≥ 2), the situation will be much more complex, with only 2-agent case finished. We show that the output of such a mechanism must satisfy a set of equations, and when p=2 the output must locate at a sphere with the segment between the two agents as the diameter. Further more, for n-agent situations, we find that the simple extension of this the 2-agent situation cannot hold when dimension m>2 and prove that the well-known general median mechanism will give an counter-example. Particularly, in L_2 (i.e., Euclidean) space with 2 agents, such a mechanism is rotation-invariant iff it is dictatorial; and such a mechanism is anonymous iff it is one of the three mechanisms in Section 4. And our tool implies that any such a mechanism has a tight lower bound of 2-approximation for maximum cost in any multi-dimensional space.
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