Improved Lower Bounds for Non-utilitarian Truthfulness

Abstract

One of the most fundamental results in the field of mechanism design states that every utilitarian social choice function admits a mechanism that truthfully implements it. In stark contrast with this finding, when one considers a non-utilitarian social choice function, it turns out that no guarantees can be made, i.e. there are non-utilitarian functions, which cannot be truthfully implemented. In light of this state of affairs, one of the most natural and intriguing objectives of research is to understand the inherent limitations in the infrastructure of truthful mechanisms for non-utilitarian social choice functions.

In this paper, we focus our attention on studying the boundaries imposed by truthfulness for two non-utilitarian multi-parameter optimization problems. The first is the workload minimization in inter-domain routing problem, and the other is the unrelated machines scheduling problem. Our main findings can be briefly summarized as follows:

• 1 We prove that any truthful deterministic mechanism, and any universal truthful randomized mechanism for the workload minimization in inter-domain routing problem cannot achieve an approximation guarantee that is better than 2. These results improve the current lower bounds of $(1 + \sqrt{5}) / 2 \approx 1.618$ and $(3 + \sqrt{5}) / 4 \approx 1.309$, which are due to Mu’alem and Schapira [SODA ’07].

• 1 We establish a lower bound of $1 + \sqrt{2} \approx 2.414$ on the achievable approximation ratio of any truthful deterministic mechanism for the unrelated machines scheduling problem when the number of machines is at least 3. This lower bound is comparable to a recent result by Christodoulou, Koutsoupias and Vidali [SODA ’07]. Nevertheless, our approach is considerably simpler, and thus may shed some new light on the core of this problem.