Social optimization problems with decentralized and selfish optimal strategies

In this paper, we consider cooperative multi-agent systems minimizing a social cost. Each agent tries to minimize the deviation from the average collective behavior and to a local command input. In principle, the optimal solution is centralized and requires a complete communication graph. We study this problem for two important system input-output norms, the induced norm and the per-agent norm squared, as function of the number of agents, n. For the case of identical agents, we show that the optimal social solution is always decentralized and characterize the local optimization problem each agent needs to solve. The solution is decentralized but not selfish in the sense that the local optimization problem is not the same as that of a single isolated agent and also depends on the number of agents. In the case of the per-agent H₂ norm squared cost, we have similar results. However, in this case, we show that the optimal decentralized selfish solution is socially optimal in the limit of large n. We study some extensions that include norm constraints, and performance indices not restricted to only penalizing the variations for averages. We also present some extensions of the results to the cases of nonuniform averaging and non uniform agent dynamics. In simple terms, these results, identify important problem classes where decentralized and possibly selfish behaviors are socially optimal, and for which inter-agent communication is unnecessary.