A pure bargaining game of dynamic cake eating

We consider a n-person pure bargaining game in which the payoff space of feasible agreements is constructed via a noncooperative cake eating differential game. At the beginning of the game the agents bargain over strategies to be played during the game while the noncooperative Nash equilibrium serves as the disagreement strategies. An initial cooperative solution (a strategy tuple) is called time consistent if renegotiating it at a later time instant yields the original solution and subgame individually rational if it remains individually rational to stick to the initial solution throughout the game. We show that time consistent and subgame individually rational bargaining solutions exist if the space of cooperative strategies is restricted to linear functions and the bargaining solution is a maximizer of some linear homogenous function.