The Communication Complexity of Payment Computation

TL;DR

This paper investigates the communication complexity of incentive compatible mechanisms, proving the exponential gap between computing only the output and the full mechanism, and providing algorithms for efficient payment computation.

Contribution

It explicitly constructs a social choice function where the communication complexity of incentive compatible mechanisms is exponentially larger, resolving an open question.

Findings

Established the exponential separation in communication complexity for incentive mechanisms.

Provided two different proofs for the exponential upper bound tightness.

Developed efficient algorithms for deterministic payment computation in key domains.

Abstract

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 $cc_{IC}(f)=\Theta(n\cdot cc(f))$, 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.