Time-Consistency in Dynamic Cooperative Games with Multiple Trajectories

This paper considers the time consistency of multistage cooperative games in which there are multiple optimal trajectories with complete information. In dynamic cooperative games, an important condition for the distribution of players’ payoffs is that individual rationality needs to be satisfied. If the players’ payoffs do not satisfy individual rationality as the game progresses along the cooperative trajectory, some players will deviate from the cooperative trajectory. If there are multiple optimal trajectories in the multistage game, the situation is further complicated by the fact that the players’ payoffs always satisfy individual rationality on some of the optimal trajectories, but not on others. Alternatively, players’ individual rationality conditions are quite different along different optimal trajectories. Therefore, four different scenarios of multi-stage cooperative games with multiple optimal trajectories are set up in the study and analysed for each of the four different situations. Four different Payoff Distribution Procedure (PDP) formulations are proposed to modify the players’ payoff distributions so that the game satisfies time-consistency on different optimal trajectories. Several theorems are given, and proofs are provided, with corresponding examples.