Nearly Complete Characterization of 2-Agent Deterministic Strategyproof Mechanisms for Single Facility Location in $L_p$ Space

We consider the problem of locating a single facility for 2 agents in $L_p$ space ($1<p<\infty$) 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 ($1<p<\infty$) 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 ($m\ge 2$), 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.