In the previous video, we defined of how to allocate the maximum joint payoff of players in multistage games. Or, we defined of what the terms of cooperative agreement can be for allocation, the joint payoff, for example, from joint venture. Let's suppose that at the beginning of the game or beginning of the process, players or economic agents agreed on some allocation, and agreed on corresponding cooperative strategy, and agreed on corresponding cooperative trajectory. Then let suppose that at some point in time, they decided to reconsider the cooperative solution, or to reconsider the terms of the cooperative agreement. Will the cooperative solution in the subgame or in this subprocess correspond to the cooperative solution of the initial game? So, will it be beneficial for the players to reconsider the terms of the cooperative agreement? In order to answer this question, we need to define the notion of cooperative subgame. We will define a cooperative subgame in each position on the cooperative trajectory x dash. In order to define the cooperative subgame, we need to define the optimization problem presented on the slide. So, for each subgame, we need to define a trajectory which maximize the maximum joint payoff of players in this particular subgame. But the part of cooperative trajectory from the initial subgame, which corresponds to the current subgame, will be optimal in this subgame? This is due to the way we calculate the cooperative trajectory. So, for each subgame, we will take the cooperative trajectory, which is a part of the initial cooperative trajectory. Then the next question is of how to allocate the cooperative payoff in this subgame along this cooperative trajectory. In order to do that, we also need to define the characteristic function of coalition S for each truncated subgame. We will do that in the same way we did that for the initial game. Then on the basis of the values of this characteristic function, we will define a set of imputations in each subgame. We will choose a cooperative solution which corresponds to the Shapley Value. On this slide, you can see the values of characteristic function for the subgame starting in the position x1 dash. So, in the subgame, which starts at the position x1 dash, along the cooperative trajectory. So, we suppose that, in the position x1 dash, so on the next step, players decide to reconsider the cooperative solution. In order to do that, on the first step, we need to calculate the characteristic function, on the next step they need to define a set of imputations which we see on the slide as well and we need to calculate a Shapley value. As you can see, the Shapley value for this subgame differs from the Shapley value of the initial game. So, does it mean that we should change the terms of cooperative agreement? Does it mean that the players would want to reconsider the agreement? We will answer on this question in the next video. On this slide, you can see a list of references, we can find more information on how to define a cooperative subgame, and how to construct a characteristic function for each cooperative subgame.