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.
翻译:我们考虑的是将2个代理商的单一设施定位在$L_p美元空间(1<p ⁇ infty$)中的问题,并给出了这种确定性战略防控机制的几乎完整的描述。我们使用一个代理商与1美元空间的设施之间的距离来表示代理商的成本。如果没有任何代理商能够通过误报其私人地址来降低成本,那么这种机制就具有战略防险性。我们显示,在2个代理商的$空间(1<p ⁇ infty$)中,一个确定性、一致的、翻译性反变异战略机制的任何位置输出必须满足一系列的等式和机制的连续和可缩缩缩缩。在一维空间中,输出必须是1个代理商的距离,这很容易在任何美元空间空间空间空间空间空间的面积(m)中(metrial2)中找到一个普通的平面机制。我们发现这种机制的输出必须满足一套方程式,当$=2的输出必须位于一个平方域, 当2的平面空间机制在2个直径之间时,这种介器的尺寸是一定的平方。