Hello. In previous section we studied the zero-sum games, when there was one player trying to maximize his payoff and there was another one acting against him or trying to minimize the payoff of the first player. The main question we wanted to answer was, of which strategies should the Colonel Blotto choose? Or of how should he allocate his regiments on two battlefields? Today, we will talk about the non-cooperative games of n players. In here, we have n agents or n players with different utilities or different payoff functions, which depend on the strategy profile which is chosen by the player. In the main question here is, of which strategy profile will be chosen by the n players? Or how to forecast the behavior of players in such a model. Consider a classical example of Prisoner's Dilemma game. This is a game of two players. Suppose that the two criminals committed a crime together and then they went to the prison and they are interrogated separately there. Each of them has two options. First option is to remain silent and the second option is to confess. If both remained silent, then they will serve only six months in prison. If both of them confess, then each of them will serve for two years in prison. But if one confesses and the other one remain silent, then the one who confesses is set free and the one who remains silent will need to serve for 10 years in prison. This is because, if someone confesses then he reveals information about the crime which was committed by the two criminals. The next classical example of the non-cooperative game is a battle of sexes. Suppose we have a husband and wife debating of where they are going to go in the evening. They also have two options. The first option is to go to the football match, which is good for a husband. The second option is to go to the theater, which is good for a wife and sometimes for a husband as well. But, if they choose different entertainments or different events they would stay at home. Suppose that they are making decisions simultaneously and independently of each other. The question is, of which strategies will they choose? How would they choose the place to go? In order to construct mathematical model, we need to introduce a notion of a non-cooperative game in normal form. It is a system Gamma which is equal to N, Xi, Ki. i from N, where N is a set of players, from 1 to n, Xi is a set of strategies of player i, and Ki is a payoff function of player i, defined on the set of all possible strategy profiles. So, the strategy profile in this class of games is a vector x1, etc... xn where xi is a strategy of a player i. So the payoff function is defined on the set of all possible strategy profiles in the game. Of course, we suppose that the players are making their moves simultaneously and independently of each other. For a case of Prisoner's Dilemma game, the set of players consists of only two players and the set of strategies consists of vectors (x1,x2) where x1 means to remain silent and x2 means confess. The same thing is true for the second player. So, the payoff function of the first player and the second player in strategy profile when both will remain silent is equal to -0.5. Why? Because they would need to spend imprison for half a year both, if they both will remain silent. The payoff function of the first and second player in strategy profile when both confess is minus two. Why? Because in case if they both confess, they would need to serve in prison for two years. In strategy profiles, when one of the players remain silent and the other one confesses, is the one who remains silent gets a payoff equal to -10. The other one who confesses gets a payoff equal to 0. Why? Because if someone confesses and someone does not, then the one who confesses is set free and the other one will need to stay in prison for 10 years. For a case of battle of sexes set of players, includes two players, husband and wife, and set of strategies also consists of vectors with two values. For a first player, it is a vector (x1,x2), where x1 is a strategy to choose to go to the football match and the second one x2 is to go to the theater. The same thing is true for the second player or a wife. The payoff function of the first player, if both choose to go to the football match is equal to 4. The payoff function for the second player if both go to the football match is equal to 1. Because in average, husband is more happier to go in a football match than the wife. For a strategy profile (x2,y2), the payoff function of the first layer is equal to 1, and the payoff function of the second layer is equal to 4. But for cases when husband and wife choose different strategies, payoff functions are equal to 0. So, they do not have any payoff because they stay at home. In order to construct a better model for this class of games, we would need to introduce the so-called Bimatrix game. So, the bimatrix game is a non-cooperative two-person gamma (N,X1,X2,K1,K2) with finite set of strategies. In our case, in case of Prisoner's Dilemma game, and in case of battle of sexes game, we have a finite set of strategies. So, for the bimatrix game, the set of players consist of two players. Each set of strategies consist of a finite set of alternatives. The payoff function of the first and the second player is defined on this finite set of alternatives and is represented by the matrix. So, the payoff of the first player is represented by the matrix and the payoff for the second layer is represented by the matrix. On the slide, you can see the bimatrix game for prisoners' dilemma game, and for each strategy profile, the payoff of the players is represented by the vector. The payoff for the first player and the payoff of the second player. For battle of sexes, the bimatrix game has the following form, as you can see on the strategy profile (x1,y1), payoffs are more than 0, and strategy profiles (x2,y2), payoffs are also more than 0, but on the strategy profiles (x2,y1) and (x1,y2), payoffs are equal to 0. Because they choose the different events for the evening. On this slide, you can see a list of references we could find more information about the non-cooperative games of N players and also can find more examples on this topic.