支付计算的通信复杂性
The Communication Complexity of Payment Computation
摘要：
设$(f,P)$为激励相容机制，其中$f$为社会选择函数，$P$为支付函数。在许多重要的设置中，$f$唯一地确定$P$（直到常数），因此常见的方法是专注于$f$的设计而忽略支付功能的作用。Fadel和Segal [JET，2009]通过从通信复杂性的角度来质疑这种方法：实现$f$（即计算输出和支付）的激励兼容机制的通信复杂性会比计算输出的通信复杂性大得多吗？即，会不会是那个$cc_{IC}(f)>>cc(f)$？Fadel和Segal表明，对于每个$f$，$cc_{IC}(f)\leq exp(cc(f))$。他们还表明，完全计算的激励兼容机制是严格困难的，比只计算输出：存在一个社会选择功能$f$，使$cc_{IC}(f)=cc(f)+1$。在后续工作中，Babaioff，Blumrosen，Naor和Schapira [EC'08]提供了一个社会选择函数$f$，使得$cc_{IC}(f)=Θ(n\cdot cc(f))$，其中$n$是参与者的数量。Fadel和Segal的指数上界是否是紧的问题仍然是开放的。在本文中，我们解决了这个问题，明确提供了一个$f$，使$cc_{IC}(f)= exp(cc(f))$。事实上，我们通过两个非常不同的证明来证明这一点。相比之下，我们表明，如果玩家是风险中性的，我们可以妥协的随机真实的预期实现（而不是确定性的事后实现）给出了$cc_{TIE}(f)=poly(n,cc(f))$的每个功能$f$，只要域的$f$是单参数或凸多参数域。我们还提供了有效的算法，在几个重要领域的确定性计算支付。
英文摘要：
Let $(f,P)$ be an incentive compatible mechanism where $f$ is the social choice function and $P$ is the payment function. In many important settings, $f$ uniquely determines $P$ (up to a constant) and therefore a common approach is to focus on the design of $f$ and neglect the role of the payment function. Fadel and Segal [JET, 2009] question this approach by taking the lenses of communication complexity: can it be that the communication complexity of an incentive compatible mechanism that implements $f$ (that is, computes both the output and the payments) is much larger than the communication complexity of computing the output? I.e., can it be that $cc_{IC}(f)>>cc(f)$? Fadel and Segal show that for every $f$, $cc_{IC}(f)\leq exp(cc(f))$. They also show that fully computing the incentive compatible mechanism is strictly harder than computing only the output: there exists a social choice function $f$ such that $cc_{IC}(f)=cc(f)+1$. In a follow-up work, Babaioff, Blumrosen, Naor, and Schapira [EC'08] provide a social choice function $f$ such that $cc_{IC}(f)=Θ(n\cdot cc(f))$, where $n$ is the number of players. The question of whether the exponential upper bound of Fadel and Segal is tight remained wide open. In this paper we solve this question by explicitly providing an $f$ such that $cc_{IC}(f)= exp(cc(f))$. In fact, we establish this via two very different proofs. In contrast, we show that if the players are risk-neutral and we can compromise on a randomized truthful-in-expectation implementation (and not on deterministic ex-post implementation) gives that $cc_{TIE}(f)=poly(n,cc(f))$ for every function $f$, as long as the domain of $f$ is single parameter or a convex multi-parameter domain. We also provide efficient algorithms for deterministic computation of payments in several important domains.
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