Power Index in Control

Abstract

Previous chapters have studied the role of non-cooperative-game approaches in the design of controllers. In contrast, this chapter discusses in studying a cooperative-game approach. Indices of power are alternative ways to solve a game, which are characterized for satisfying a certain system of axioms. Some of these power indices are, among others, the Shapley value, the Banzhaf–Coleman index, or the dictatorial index (Owen, Shapley, Int J Game Theory 18(3):339–356, 1989, [1]). Moreover, even though for some examples different indices are equivalent, in the general case they adopt different values. Specifically, this chapter addresses the role of the Shapley power index in the design of controllers. In Owen, Shapley (Int J Game Theory 18(3):339–356, 1989, [1]), it has been shown that the computation of the Shapley value is more complex than the computation of other power indices. The main reason is that the computation of the Shapley value involves a combinatorial explosion since it evaluates all the possible coalitions that can be made among players. Therefore, this fact implies a high computational burden. In addition, the computation requires information from all the players, i.e., it is computed under centralized communication structures, which constitutes an additional challenge for real applications, specially in cases with large amount of players, where big communication networks would be needed.