When Selfish is Socially Optimal

In this article, we consider cooperative multiagent systems, where independent agents need to minimize a collective social cost. Given the coupling in the cost, the optimal strategy requires interaction and communications among all the agents in general. In this article, we unveil important classes of social costs which are optimized by decentralized and selfish solutions, hence eliminating the need for an interagent communication network. In particular, we focus on a set of n independent agents coupled only through an overall cost that penalizes the divergence of each agent from the average collective behavior. Adopting input-output methods, we show that optimal decentralized and selfish solutions are possible in a variety of standard input-output cost criteria. These include the cases of ℓ 1, ℓ 2, ℓinduced, and H2 norms for any finite n. Moreover, if the cost includes nondeviation from average variables, the above results hold true as well for ℓ 1, ℓ 2, ℓ induced norms, and any n, while they hold true for the normalized, per-agent square H2 norm, cost as n. The results of the article demonstrate that selfish behavior can be socially optimal in nontrivial cases. We further expand our analysis to more general controller structures that induce collective tracking of exogenous common signals.