In the previous video, we managed to construct a game theoretical models for Battle of Sexes and Prisoner's Dilemma. But we did not answer on the main question of what will be the principle of optimality in this games or how can we forecast the behavior of players in this game. In order to do that, we need to introduce an approach, and the classical and widespread approach is the Nash equilibrium. The Nash Equilibrium is a strategy profile from which for both players the individual deviation is not beneficial. Later on, we will talk about the good properties of the Nash equilibrium and why it is so popular. Of course nowadays there are other optimality principles with also good qualities. But this one is the first principle of optimality for non-cooperative games in normal form, and this is the most popular one. So, the strategy profile x* is called the Nash Equilibrium if the following condition is satisfied, it means that the payoff of player i K_i(x*) in strategy profile x* or Nash equilibrium is more or equal to the payoff of a player i if all players but the player i use strategies from the strategy profile x*, and the player i chooses an arbitrary strategy x_i from set of strategies X_i. This condition should hold for all players from 1 to n. Let's try to compute a Nash equilibrium for Prisoner's dilemma game. So, the bimatrix game is presented on the slide but the question is how to find it. But, since we have only four strategy profiles, then the only thing we need to do now is only to check this condition for each of this strategy profiles. So, for each strategy profile, we need to check two conditions and if we do that, we can say that the strategy profile (x2,y2) is actually a Nash Equilibrium. How we can check that, let's check the following conditions which are also presented on the slide. So, the condition for the first player has the following form, so we need to check if the payoff in strategy profile (x2,y2) for the first player is more or equal to his payoff when he deviates. We have only one option for deviation so he can only choose a strategy x1, and when the second player chooses a strategy from the strategy profile (x2,y2) which is y2. Here we can see that the -2 is more than -10 which means that this condition is satisfied. Then we also need to check the condition for the second player. In case of a second player, we need to check if his payoff in strategy profile (x2,y2) which is -2 also more or equal to his payoff when first player chooses a strategy x2 from Nash equilibrium and the second player deviates. He has the only one option for that which means that he will choose a strategy y1, and you can see that the condition also holds. But as you can see, this particular outcome or this strategy profile is not beneficial for the players or it is not a good outcome for the criminals. Of course it would be better for them not to say confess, confess but to remain silent for both of the players because in this particular situation they would only serve in prison for half a year. But the problem here is that they do not communicate while they're making the decisions, they are interrogated independently of each other and simultaneously. It means that each one of them do not know even if they would choose a strategy profile (x1,y1), and they would communicate before then but after each of these prisoners does not know if the other one will stick to this agreement. Because as you can see if both of them chooses the strategy profile (x1,y1), and at some point if the second player deviates, he would receive a better payoff, so why not to use that? But the strategy profile (x1,y1), will be considered also as a good solution but for the other principle of optimality which we will talk about later. For a Battle of Sexes, the bimatrix game is presented on the slide, and in this case we will have two Nash equilibriums. So, we have two strategy profiles which are good for both players in this simultaneously decision-making process. These are the strategy profiles (x1,y1) and (x2,y2). We can easily check that, let's do that for one strategy profile then you can do it in the same way for the second one. So, for the strategy profile (x1,y1), payoff of the first player is equal to 4 and we need to check that it should be more equal to his payoff when he deviates or chooses a strategy x2 and the second one sticks to the Nash equilibrium. In this case 4 is more equal to 0, then for the first player condition holds then. For the second player, we also need to check the condition if payoff of the second player in this strategy profile, which is equal to 1, more or equal to his payoff when he deviates or chooses a strategy y2, and the first one sticks to the Nash equilibrium. In this case you can see that this condition still holds, so 1 is more or equal to 0. In the same way we can do for the second strategy profile. But the main question still stands of which strategy profile will this both players choose because both of them are good but where would the husband and wife go, to the football match or to the theater? But we did not answer on this question at the moment. Next principle of optimality that we will consider is called the Pareto optimal solution. The pareto optimal solution x dash is solution for which there is not exist another strategy profile where payoffs for both players or for all players are better, or the following two conditions are satisfied. So, for each player i payoff in some strategy profile x is more or equal to the payoff of player i in strategy profile x dash. There exists at least one player i0 for which his payoff in strategy profile x is strictly more than his payoff in strategy profile x dash. For Prisoner's dilemma game, there are four strategy profiles and actually three of them are Pareto optimal. The first one (x1,y1), is Pareto optimal there because they're not exist another strategy profile which is better for both players. Accidentally, strategy profiles (x2,y1) and (x1,y2) also are Pareto optimal because for this case the payoff of the second player and the first player correspondingly is equal to 0, and there are no strategy profiles where these payoffs are better or even also equal to 0. On this slide, you can see a list of reference where you can find more information about the Nash equilibrium and Pareto optimality principles, and also you could consider more examples.