In the previous video, we introduced the notion of Nash Equilibrium. This is a solution for non-cooperative games, which is a very classical and a widespread solution. But why is that so? Because of the two things. So, the first thing its physical meaning. It is a solution for which the individual deviation for each of the player is not beneficial. So, it can be used as a forecasting tool for the processes. The second thing and the most important thing is that, this solution exist for each finite non-cooperative game. We will talk about this in this video. But in order to do that, we need to introduce a notion of a mixed strategies. We will do that in the same way as we did for a zero-sum games. We say that the mixed strategy of a player one and two correspondingly is a vector which assigns for each pure strategy of player one for example, a probability that this strategy will be chosen. So, for the first player, mixed strategy is a vector x equal to ksi 1, etc... ksi m. Where ksi i is the probability of choosing the pure strategy i. Ksi i is more equal to zero and the sum of all values of the vector ksi_i is equal to one. The same thing is true for the second player. So, the mixed strategy for the second player is vector y, which is eta1, etc... eta n. The sum is equal to one and each value is more equal to zero. But the question is of how these strategies can be realized? They can be realized in the same way as we studied for the zero-sum games. So, the first thing we do is, we define the probability distribution on the set of pure strategies for each player. Then on the next step, we can use a table of random variables in order to generate a certain pure strategy which corresponds to this probability distribution. So, for our case of Prisoner's dilemma game, we could say that the mixed strategies of a first prisoner and the second prisoners are the following vectors 0.8 and 0.2, and 0.5 and 0.5. This would mean that, the first strategy profile (x1,y1) which is remains silent, remains silent is chosen with probability 0.2 multiplied by 0.5. For battle of sexes, we could use the following mixed strategies 0.2, 0.8, and 0.8, 0.2. This would also mean that each strategy profile is chosen with probability which is equal to the probability of choosing the corresponding pure strategies. So, for the strategy profile x1, y1, it is a 0.2 multiplied by 0.8. The next thing we need to do is to define a payoff function in mixed strategies. We will define it as a mathematical expectation of payoffs given that the players use mixed strategies or define the probabilities that each particular pure strategy will be chosen. The corresponding formulas are presented on the slide. Let's consider a mixed strategy 0.2, 0.8 and 0.8, 0.2 for a battle of sexes game. Then the corresponding payoffs of players are calculated in the way it is presented on the slide. Very important thing about the Nash Equilibrium in the non-cooperative games is that, it exist in mixed strategies. So, in any finite n-person game Gamma in normal form, there exists at least one Nash Equilibrium in mixed strategies. This is a fundamental result for non-cooperative games, and it has proved using the theorem of Kakutani presented below. Next result, which is presented on the slide, can be used in order to determine if the given strategy profile x* and y* is actually a Nash Equilibrium. So, it turns out that we only need to check a finite number of conditions for a strategy profile to be Nash Equilibrium. Before we did that for a Bimatrix game but we only worked with pure strategies. But now as it turns out, if we have a strategy profile or Nash Equilibrium in mixed strategies, then we only need for the first player to check the condition when a payoff of the first player is more equal to payoff for the first player when the second player uses a mixed strategy y* from Nash Equilibrium and the first player uses the pure strategy. So, we only need to check the condition in the definition of Nash Equilibrium only for pure strategies. The same is true for the second player. On this slide, you can see the result which can be used in order to find the Nash equilibrium in a simple bimatrix game. This result says that, "If x* and y* is a Nash Equilibrium, then in case of a first player, the following condition is satisfied. The payoff of the first player in x* and y* is equal to the payoff of the first player when the second one uses a mixed strategy y* from Nash equilibrium, and the first one uses a pure strategy xi from the spectrum of a mixed strategy x*." The spectrum of a mixed strategy is a set of pure strategies for which the probabilities that they will be chosen in x* is strictly more than zero. So, a set of pure strategies that could be realized given that, we use a mixed strategy x*. The same condition holds for the second player. Let's see how we can use this conditions in order to solve a simple bimatrix game. Let us try to apply the results obtained in the previous slide for a game Battle of Sexes and find the Nash Equilibrium in mixed strategies. In order to do that, we need to write the two equations which are present on the slide. The first equation says that, The payoff of the first player given that the first player uses the mixed strategy x* and the second player uses a pure surgery y1 is equal to the payoff of the first player when the first player uses mixed strategy x* and the second player uses a pure strategy y2. This equation follows from the first condition from the theorem presented on the previous slide. The second equation follows from the second condition and we will consider that later. For the first equation, we can rewrite the left and the right-hand side as a functions of a mixed strategy of the first player. Therefore, we can derive the mixed strategies of the first player from this equation. We will get the equation for ksi* equal to 1-ksi*. As a result, we can write that the mixed strategy of the first player from the Nash Equilibrium x* is equal to ksi*, 1-ksi* is equal to 0.8, 0.2. For a second player, the equation says that, The payoff function of the second player given that the second player uses a mixed strategy y* from Nash Equilibrium, and the first player uses a second pure strategy equal to the payoff of the second player when the second player uses a mixed strategy y* from Nash Equilibrium, and the first player uses a first pure strategy. From this equation, we can define the equation for a mixed strategy of the second player, which is at eta* equal to 4-4eta*. As a result, y* equal to eta*, 1-eta* equal to 0.2 and 0.8. The payoff for both players in this particular strategy profile or in Nash Equilibrium is equal to 0.8. On this slide, you can see a list of references. You can find the proof of the fundamental theorem for the non-cooperative games, the theorem of existence the Nash Equilibrium in mixed strategies. Also, you could find the proofs of the results connected to the algorithm for finding the Nash Equilibrium in Bimatrix games.