A bounded-risk mechanism for the kidney exchange game

In this paper, we introduce and study the notion of low risk mechanisms. Intuitively, we say a mechanism is a low risk mechanism if the randomization of the mechanism does not affect the utility of agents by a lot. Specifically, we desire to design mechanisms in which the variances of the utility of agents are small. Inspired by this work, later, Procaccia et al. (0000) study the approximation–variance tradeoffs in mechanism design. In particular, here we present a low risk mechanism for the pairwise kidney exchange game. This game naturally appears in situations that some service providers benefit from pairwise allocations on a network, such as the kidney exchanges between hospitals. Ashlagi et al. (2013) present a 2-approximation randomized truthful mechanism for this problem. This is the best-known result in this setting with multiple players. However, we note that the variance of the utility of an agent in this mechanism may be as large as Ω(n²), where n is the number of vertices. Indeed, this is not desirable in a real application. In this paper, we resolve this issue by providing a 2-approximation randomized truthful mechanism in which the variance of the utility of each agent is at most 2+ϵ. As a side result, we apply our technique to design a deterministic mechanism such that, if an agent deviates from the mechanism, she does not gain more than 2⌈log₂m⌉ where m is the number of players.