In the previous video, we defined the notion of cooperative subgame. We defined a cooperative subgame for each position on the cooperative trajectory. We managed to calculate the cooperative solution or the Shapley value for some subgame and then we saw that that solution does not correspond to the Shapley value of the initial game, which means that the players would want to reconsider the terms of the cooperative agreement. But how to construct a cooperative solution or how to construct the parameters of cooperative agreement so that at any point on the cooperative trajectory or any point in time, the players would not want to reconsider the cooperative agreement. In order to do that, we need to define a special payments or the imputation distribution procedure. So, we will define of how can we allocate the imputation which corresponds to the player i, for example, ksi_i in time. The imputation distribution procedure we will define as a matrix Beta, where each element of this matrix or element Beta_ik corresponds to the payment that the player i would receive on the stage k. The sum of the elements of the row number i of the matrix Beta is equal to ksi_i. So, if a player has decided to allocate the joint payoff according to the Shapley value at the beginning of the game and then they use some imputation distribution procedure, so, somehow they decide to allocate the payments for each player in the game, then the sum of the payments along the cooperative trajectory would be equal to the Shapley value of the initial game. On this slide, you can see the Shapley value for the initial game we considered before and some imputation distribution procedure. As you can see, all three players receive payments on each stage of the game or in each position along the cooperative trajectory. But what happens if we define the imputation procedure in the way that all players would receive all payoffs on the first stage? Then, on the stage two and then the stage three for the players, the cooperation will no longer be profitable. They will want to deviate from the cooperation. So, how can we define the imputation distribution procedure so that the renegotiation of the terms of the cooperative agreement will not be profitable for them? In order to do that, we need to define the notion of time consistency. We say that the cooperative solution or a set of imputations is called time-consistent, if for each imputation from this set, the corresponding IDP satisfies the following system of linear equations. Let us consider the first equation. The first equation means that if we define the Shapley value for the initial game model, and then we define some IDP and make a payment on the first stage to all of the players, then on the next stage we recalculate the Shapley value in the subgame then the payment on the first stage should be equal to the Shapley value of the initial game minus Shapley value of the subgame. So, it means that after the first payment the payoff that we still need to obtain is exactly the same as the Shapley value in the subgame. It means that there is no need to renegotiate, because we will have the same payoff at the end, and the same is true for each subgame. Let's go back to our example of signing a package of documents and here on the slide you can see the way of how we can calculate the IDP so that the cooperation solution will be time-consistent. What we need to do is to calculate the Shapley value of the initial game or in game starting at the position X0, then we need to calculate the Shapley value starting at the position X1 dash and then calculate the IDP for a first stage. For the second stage, the IDP is equal to the Shapley value in the game starting at position X1 dash minus Shapley value in the subgame starting at the position X2 dash and so on and so on. But as you can see some of the values of the IDP are negative. It means that we do not make a payment but for example we freeze the money on the bank account. In some cases, this procedure cannot be implemented. But this procedure will give us the time-consistent solution. There is another version of the time-consistency called the strong time-consistency. We say that the cooperative solution or the set of imputation is called a strong time-consistent if for any imputation from the solution there exist an IDP that satisfies the following condition. For example, in the beginning of a game player decide to choose a core as a cooperative solution and they picked one of the imputations from the core, and calculate the corresponding IDP. Then, according to this IDP, they would receive some payments till the time instant t. Then, for example, at a time instant t, they decided to renegotiate or to recalculate the cooperative solution namely the core. So, as a result they would receive a big set of new imputations in the subgame. The strong time-consistency means that the payments that we received on the interval from t0 to t plus all possible payments that we receive from the core from the subgame should be the subset of the initial core. So, no matter when we recalculate the cooperative solution or we renegotiate the cooperative agreement terms we will have some solution from the initial set. So, there is no need to renegotiate the cooperative agreement or to recalculate the cooperative solution in any subgame along the cooperative solution. On this slide, you can see list of references where you can find more information on the imputation distribution procedure, time-consistency property and a strong time-consistent property.