Hello. Today, we will consider two topics. The first one is bargaining problems. The second one is cooperative games in characteristic function form. So, the first topic is used for modeling the cooperative or strategic agreements between players or economic agents. It is used when we want to define which strategy should we choose in order to sign a cooperative agreement. The second topic is used for modelling of how can we allocate payoff which is obtained if we cooperate. So, for example we choose some cooperative strategies, for example, maximization of joint utilities or payoffs, then the question stands of how can we allocate this payoff among players so that the cooperation would be beneficial for all of the players? Now, let's start with bargaining problems and some classical solution for that which is called the Nash bargaining solution. Let's start with classical example from game theory called the Battle of Sexes. In here we have husband and wife choosing of where they want to go at the evening. They have two options. The first option is to go to the football match. The second option is to go to the theater. Of course, the couple wants to spend the evening together. They do not want to be separated. As we already discussed before, this problem can be modeled using the bimatrix game which is shown on the slide. Here, the first player is the husband. The second player is wife. The husband has two pure strategies x_1 and x_2 and wife also has two pure strategies y_1 and y_2. Here, x_1 is when husband chooses to go to the football match and x_2 is the pure strategy when husband chooses to go to the theater. The same is true for the wife. Let's consider some strategy profile. So, the first strategy profile is (x_1,y_1). Here the payoffs are four and one. So, the husband receives payoff equal to 4 and wife receives payoff equal to 1, if both of them choose to go to the football match. If both of them choose to go to the theater, which is the strategy profile (x_2,y_2), then the payoffs are one and four correspondingly. If we consider other strategy profiles which are (x_2,y_1) and (x_1,y_2) then the payoffs there are equal to 0 because both of the couple doesn't want to be separated. In this particular bimatrix game, the Nash equilibrium are strategy profiles (x_1,y_1) and (x_2,y_2) and we also have the Nash equilibrium in mixed strategies. The Nash equilibrium here can be used for forecasting of how this process would develop or what strategies will husband and wife choose if they would decide independently and simultaneously. But what happens if they would want to make an agreement or if they would want to bargain of where do they want to go? For that, we need to consider the set of all possible outcomes. By the outcome, we understand a vector of payoffs. On this slide, you can see the set Rab which is the set of all possible outcomes. Points a, b and R are the payoffs of the players in pure strategies. Point d is the payoff of players in Nash equilibrium within mixed strategies. Point d is also called the disagreement point, but we will talk about that later. So, we suppose that the players do not want to make an agreement of something which is less than the Nash Equilibrium. So, they say that we want to have a good bargain here. Therefore, as the bargaining set or the set of payoffs or outcomes where we want to choose the solution is defined as follows: e, d, c, b and a. So, what is the bargaining problem? The bargaining problem is problem when we need to define a point from bargaining set which is a function of S - bargaining set and disagreement point d. The disagreement point can be defined in different ways. It could be a Nash equilibrium. It could be defined using maximin approach or it could be even defined manually. But in our case, we will use the Nash equilibrium approach. Let's go back to our example the Battle of Sexes. In here, the set of player is one and two. So, we have two players. Bargaining set was shown on the slide before and the disagreement point is the payoffs in Nash equilibrium. Now, the question stands of which point from the bargaining set do we want to choose? For that, there are several classical approaches. One of them and the most famous one is the Nash bargaining solution. The Nash bargaining solution can be used for n-player games or for the game with arbitrary number of players. But in our case, we have two player game. Nash bargaining solution and any other bargaining solution or cooperative solution in game theory is defined using the set of axioms. What is the axiom? The axiom is a property, a mathematical property which has a certain physical meaning. When we define cooperative solution, we want it to be for example, Pareto-optimal. So, we want it to has a good outcomes. We also want to have other properties. So, we define a set of properties or set of axioms. Then we derive the solution and then we try to construct the analytical form or explicit formula for this solution. So, for the Nash bargaining solution, we define four axioms. The first one is Pareto-optimality. So, the bargaining solution in our case is a vector and it is Pareto-optimal when there is no vector which is better than the chosen one for any vector value. So, we say if there is no vector which is better for any vector value, then we say that the chosen solution is Pareto-optimal. Then the second axiom is called symmetry. We say that the solution satisfies axiom of symmetry if we will have the same solution when we change the number of players, or we change the names of the players. So, solution only depends on some values of payoffs. So the third axiom is called scale invariance. It says that if we change the bargaining set using some linear operator, for example, we expand the bargaining set by multiplication or constant or something like that. Then what we will get is that we could calculate the bargaining solution using the initial set and the expanded set, then we can calculate the bargaining solution on the expanded set as the bargaining solution with linear operator in the initial set. That's it. The fourth axiom is called the independence of irrelevant alternatives. It says that if we consider the initial bargaining set, then we define the specific bargaining solution. Then we can consider the subset of the initial bargaining set which includes the corresponding bargaining solution. Then if we recalculate some bargaining solution in the subset of the initial bargaining set, then it will be equal to the bargaining solution in the initial bargaining set. So, it means that the bargaining solution defined using the initial bargaining set only depends on the alternatives or the outcomes which are relevant. It was proved that there exists a unique function satisfies the axioms which were defined on the previous slide and it can be calculated using the following explicit formula. The formula seems to be pretty easy to compute, but as we increase the number of players, the computation time will be larger. Also the solution that satisfies these four axioms and as it turns out can be calculated using this formula, we will call the Nash bargaining solution. Let's calculate the Nash bargaining solution for our example of Battle of Sexes. On the slide, you can see that the Nash bargaining solution is located right in the middle of the interval a and b. This is only because this interval a and b which actually is the set of Pareto-optimal outcomes is symmetrical to the disagreement point d. On this slide you can see a list of references where you can study of how we can prove the theory which corresponds to the Nash bargaining solution and also you can find there other examples.