Balancing Asymptotic and Transient Efficiency Guarantees in Set Covering Games

Game theoretic approaches have gained traction as robust methodologies for designing distributed local algorithms that induce a desired overall system configuration in multi-agent settings. However, much of the emphasis in these approaches is on providing asymptotic guarantees on the performance of a network of agents, and there is a gap in the study of efficiency guarantees along transients of these distributed algorithms. Therefore, in this paper, we study the transient efficiency guarantees of a natural game-theoretic algorithm in the class of set covering games, which have been used to model a variety of applications. Our main results characterize the optimal utility design that maximizes the guaranteed efficiency along the transient of the natural dynamics. Furthermore, we characterize the Pareto-optimal frontier with regards to guaranteed efficiency in the transient and the asymptote under a class of game-theoretic designs. Surprisingly, we show that there exists an extreme trade-off between the long-term and short-term guarantees in that an asymptotically optimal game-theoretic design can perform arbitrarily bad in the transient.