On the regularization of a cooperative solution in a multistage game with random time horizon

In this paper, we consider a general class of cooperative multistage games with random time horizon and discuss the problem of implementing a cooperative solution. It is known that in many cases a cooperative solution can be time-inconsistent and hence not realizable. To solve this problem, the imputation distribution procedure was proposed. However, the computed payment distribution scheme may result in negative payments which are not feasible. In this case, one has to carry out a regularization procedure as described in the paper. We describe a general regularization scheme and apply it both to the core and to the Shapley value. It is shown that for the mentioned two cases the regularization can be carried out in two alternative ways thus providing a basis for developing efficient numerical schemes. For the Shapley value the regularization procedure was elaborated and described in the form of an algorithm. The obtained results are illustrated with two numerical examples. (C) 2018 Elsevier B.V. All rights reserved.