 So far, we have discussed various types of properties and various results involving those properties. So let us do a quick recap. We have discussed iterated elimination of dominated strategies in order to find some equilibrium of the game. Then we have talked about preservation of equilibrium when we do this kind of iterated elimination. We have defined the stability notion by pure strategy Nash equilibrium and security notion by min max, max min strategies. And we have shown that these two properties, these two notions coincide in the case of this special game called the matrix games. And then we have also looked at certain kind of games where pure strategies may not exist. Pure strategy Nash equilibrium may not exist. And one such game is given in this figure where we cannot find a pure strategy Nash equilibrium. So that is one reason of refining the equilibrium notion as we have done in the past. When we could not find dominant strategy equilibrium, we went for the pure strategy Nash equilibrium. Now it's time to weaken it even further and therefore we might get some other equilibrium strategy. And also from an application point of view, if you think about the first example that we have given for the neighboring kingdoms dilemma, we have seen that we just made an assumption that the kingdoms can either invest entirely on agriculture or on warfare. That was for simplicity. But in practice, there are cases where the whole resource of a specific kingdom can be divided maybe equally between these two actions, agriculture and warfare, or maybe one third and two third. How should we represent that in the form of a game? So to denote that, we now come to the world of mixed strategies. So as the name suggests informally, this is a probability distribution over the strategies. So instead of picking the strategies as pure, we are just mixing them and we are doing this mixing in a probabilistic way. So in the same example, we can say that the player one is mixing these two strategies L and R in this proportion two third and one third. So with probability two third, it will pick L and with probability one third, it will pick R. Similarly, the other player will pick these two strategies with probability four by five and one by five. Alright, so let us make our notation a little more clear to formally represent all these things. So suppose we have a finite set A, then we are going to define this delta of A as the probability distributions on this set A. So what does that mean? So we have a vector. So P is a vector. So suppose A has three elements, let's say A1, A2, A3. Therefore P will be nothing but a vector of length three each representing each of these elements in A and they should sum to one. So let's say 0.2, 0.3 and 0.5 and 0.3. So that is one valid entry in this set delta of A, which is the set of all probability distributions over A. Now what is a mixed strategy for player I? So sigma I with which we are going to denote the mixed strategy of player I is a member of delta of Si. So Si in this case is the finite set and what we are going to say is that the agent I is putting certain probabilities on all these strategies that is available to it. And we are doing it in such a way that the sum, this is always non-negative and it sums to one. Very simple definition and because we are discussing non-cooperative games that no player is actually talking to each other and sharing information before taking their decisions, they choose their strategies independently and therefore the mixed strategies are also chosen independently. So all these probability distributions are actually independent of each other. So when we are talking about the joint probability distribution of player I picking S1 and player II is picking S2, that is just given by the product of the individual marginals. Similarly, these are all analogous definitions. So when we are using the utility of a player I at a mixed strategy profile, sigma I and sigma minus I and sigma I is nothing but one element in that delta of capital Si. And similarly for all the other players, if you list out their strategies, their mixed strategies, that vector of mixed strategy profile except player I is denoted by sigma of minus I. So this is the same notation as before. So how do we define, how do we write the utility of player I when the players are picking their mixed strategies? So what it means is that we first look at the utility at a pure strategy profile S1 to Sn and multiply that with the corresponding probability of picking that strategy profile. So because these are all independent, this will just be products and then you are going to sum over all possible strategies of all these players. So therefore you are getting an expected utility for all the strategy profiles where the expectation is taken with respect to this probability distribution according to their mixed strategies. So in this case, we are just overloading this notation. So utility of I for the pure strategies is also denoted by UI and we are also denoting the same utility I to denote the mixed strategies. Ideally it should have been different. We should have used different notations but because these are two different contexts and from the context you will be able to figure that out. So there is no need and in literature it is not really distinguished. So in this case I have explicitly written the whole utility expression just to make sure that you understand the whole expression how the utility at a mixed strategy is computed. So said as we mentioned before utility at a mixed strategy is the expectation of the utilities at the pure strategies. And so all the rules of all the known laws of expectation holds for instance linearity holds. So we will make use of this fact sometime later but this is just for an information. So what is an example of a mixed strategy and let me illustrate with an example what is the utility at a specific mixed strategy profile. So let us look at the same example that we have shown before. So the game is the penalty shootout game but now the players are not picking pure strategies but they are picking mixed strategies. Player one is picking this two third and one third and player two is picking four fifth and one fifth. So if you are looking at the utility of player one so we are only concerned about the utilities of player one in different strategy profiles. We are first going to write down the utility so let us say the first utility here for player one is minus one and we are going to multiply that with the probability of occurrence of this strategy profile. So L comma L can now happen with probability two third times four fifth because that is the way player two and player one can pick this strategy. Similarly if you look at one then you are going to multiply two third and one fifth and similarly you finish up one third times four fifth and one third time one fifth and multiply them with the corresponding utilities. This is the expected utility when player one is playing this particular strategy so sigma one is given by two third and one third and sigma two is given by four fifth and one fifth. Similarly you can write down the utility for player two where this numbers will get replaced by the second entries here. You can also talk about mixtures of mixed strategies that two mixed strategies are being mixed but then that will also give rise to a new mixed strategy and this kind of a notion is rarely used in practice but it is good to know for your information.