Subgame Consistent Cooperative Solution in Differential Games

Abstract

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games which ensures that the extension of the solution policy to a later starting time and any possible state brought about by prior optimal behavior of the players would remain optimal. In many game situations payoff (or utility) of players may not be transferable. It is well known that utility in economic study is assumed to be non-transferrable or comparable among economic agents. The Nash (1950, 1953) bargaining solution is a solution for non-transferable payoff cooperative games. Strategic interactions involving national security, social issues and political gains fall into the category of non-transferrable utility/payoff (NTU) games. In the case when payoffs are nontransferable, transfer payments cannot be made and subgame consistent solution mechanism becomes extremely complicated. In this Chapter, the issue of subgame consistency in cooperative stochastic differential games with nontransferable payoffs or utility is presented. In particular, the Chapter is an integrated exposition of the works in Yeung and Petrosyan (2005) and Yeung et al. (2007). The Chapter is organized as follows. The formulation of non-transferrable utility cooperative stochastic differential games, the corresponding Pareto optimal state trajectories and individual player’s payoffs under cooperation are provided in Sect. 6.1. The notion of subgame consistency in NTU cooperative stochastic differential games under time invariant payoff weights is examined in Sect. 6.2. In Section 6.3, a class of cooperative stochastic differential games with nontransferable payoffs is developed to illustrate the derivation of subgame consistent solutions. Subgame consistent cooperative solutions of the class of NTU games developed in Sect. 6.3 are investigated in Sect. 6.4. Numerical delineations of the solutions presented in Sect. 6.4 are given in Sect. 6.5. An analysis on infinite horizon NTU cooperative stochastic differential games is provided in Sect. 6.6. A chapter appendices containing proofs are given in Sect. 6.7. Chapter notes are given Sect. 6.8 and problems in Sect. 6.9.