 We have used the MSNE characterization theorem to find an algorithm to find MSNE and in this module we are going to prove that fundamental theorem which actually gave rise to that algorithm. So, let us reiterate what was this MSNE characterization theorem. It says that a mixed strategy profile sigma i star sigma minus i star is a mixed strategy in Nash equilibrium if and only if these two conditions hold for every player. The utility of that player when it plays the pure strategy Si and the other players pick the mixed strategy sigma minus i star is going to be the same for all these strategies in the support of sigma i star. And the second condition was that this utility is going to be at least as much as the utility any other strategy which is not in the support. So, let us try to prove that. In the proof we will need two results which I will state as observations. These are very simple observations about convex sets and its properties. So, the first thing is that we are going to find the maxima over all the possible mixed strategies of player i. So, what we are saying is that find the maxima over all mixed strategies. So, I can write this expression is this utility expression as follows. So, this is the utility at Si, so strategy pure strategy Si and the other players are picking the mixed strategy sigma minus i and what I am doing is I am looking at sigma i Si and taking it sum over all Si in capital Si. This is what this utility means. Now we are trying to maximize it. Now think about this problem we are trying. So, sigma i Si we know that those are probability distribution. So, this utility for Si there are let us say some k number of strategies in the strategy set of player i. So, all of this individual Si living in that set will give some utility for this player and now we have this k numbers and we are mixing them taking a convex combination of those numbers and we are trying to find out which convex combination will give you the maximum. Do you think of any convex combination which gives you the maximum? It is the maximum value among those k numbers where you can put all the mass and that will give you the maximum convex combination and that is exactly what is written on the right hand side. If you just pick that strategy which gives you the maximum, so this max is nothing but the maximum over all those k numbers that is going to be the maximum value here. No matter in whatever way you want to find this maxima if you try to find out the probability distribution that maximizes this expected utility of this player that is equal to the case where you are putting all the mass on the maximum value. So, therefore this maximum value should be same. So, this is the first observation. This is a very simple observation yet quite important and interesting. The second thing is that let us look at the quantity. So, again the left hand side is the same, but we have replaced only the strategy of the other players as sigma minus i star. So, what is sigma i star? So, sigma i star and sigma minus i star is the mixed strategy Nash equilibrium. So, what is from the definition of it? What do we know? We know that if we use this, this is going to be the maximum over all the other strategies or over all the other mixed strategies. Now, we have already seen this equality to hold from the first observation. Now, what we are going to argue is that this is equal to the maximum where you are looking at the strategies not on the whole set of Si, but just the support of sigma i star. And why is that true? So, suppose this is not true. So, suppose this maxima is not coming from this set from the support, but it is living somewhere outside. So, let us say there exists some Si prime which is not in the support and that has the maximum value. Then what we can say is that we can put the entire probability mass on that strategy profile, on that strategy of that player i. And that because that is going to be the maximum that is certainly is going to give you a higher utility than the expected utility of sigma i star. Because when you are looking at this sigma i star, you do not have the maxima there. All the numbers that you have on the support, all the utilities that you have for the strategies on the support, even if you take the convex combination of that, then also whatever you are going to get at max is the maximum value there. But you already have some other strategies i prime where you have a strictly larger utility value. So, if I put, so this is a different mix strategy, of course, it is a pure strategy. But yes, of course, that is a degenerate mix strategy where I am putting all the probability mass to that Si prime and that is going to give you strictly larger utility, expected utility than the expected utility that you can get on the strategy profile sigma i star sigma minus i star. So, that contradicts the fact that sigma i star, sigma minus i star is a mixed strategy Nash equilibrium. So, therefore, that cannot happen. So, whatever we have assumed is wrong. So, therefore, max must leave within the support of sigma i star. Okay. So, with these two observations, we are now in a position to prove this result. So, there are two parts of this proof because it is a necessary and sufficient condition. First, we will prove the forward direction. That means we will start with assuming that sigma i star sigma minus i star is mixed strategy Nash equilibrium and show these two conditions hold. And in the reverse direction, we will assume these two conditions to hold and show that sigma i star sigma minus i star is a mixed strategy Nash equilibrium. So, the first direction is a forward direction. So, here we are given this is a mixed strategy Nash equilibrium. And we have just used, we are just going to use the definition of mixed strategy Nash equilibrium. So, this is by the definition we are maximizing with respect to sigma i, all the sigma i's. Now, we make use of the first observation that it is going to be the max over that strategy space. So, maximizing the convex combination is equivalent to maximizing with respect to the strategy, the pure strategy. And now we also know from that second observation that this is going to be the maximum over the support of sigma i star. So, let us call this the first observation, first subresult. Now, also we know by definition of the expected utility, you can write this utility at the mixed strategy profile in this form, the expected utility form. Now, because this the support, the definition of this support is that beyond this support, all this for all the strategies sigma i star has zero probability mass. So, I can just remove those strategies from this summation. And I can just reduce the summation to be the support of sigma i star. Now, I know that this first subresult and the second subresult are equal because the left hand side is the same for both these things. Now, what is this two things saying? It is saying that the max, so this one, so you are looking at the same set of delta sigma i star, the support and the max is exactly equal to the positive weighted average. So, notice this term, it is a positive weighted average. So, all these terms now here is positive. So, there is no zero number there. When can you take a convex combination of all these numbers which is equal to the max of those numbers? It can happen, think about it, it can happen only when all the numbers are equal. So, there is no other way. If there is a, there are dissimilar numbers, one is slightly larger than the other numbers, then the mix of that, the convex combination of that will certainly be the, certainly be smaller than that maximum number. So, it can never happen unless your, all the numbers are exactly equal. And that is exactly what we are going to conclude from here. When all the values are same, that means all this ui si sigma minus i star are going to be the same for all the strategies which is living in this support of sigma i star. So, that essentially proves the first condition, the condition 1. Now, let us go to the condition 2. So, how can we prove condition 2? So, what condition 2 is saying is that for all the strategies which are in support, the utility at that strategy is going to be large, at least as much as the utility for a strategy which is not in the support. So, assume for contradiction that is not true. So, there exists some strategy which is in the support and some strategy si prime which is not in the support, such that the inequality is flipped. So, then again we can use the very similar argument because now we have a competing probability mass. So, we have a strategy of si prime where the utility is going to be larger, then the probability mass that I was earlier giving. So, consider the strategy. So, sigma i star is giving different probability masses to different strategies. So, there is si and there are other strategies also in the support of sigma i star for player i. So, because this si prime has a larger utility now, so whatever probability mass you are giving on this si, you just shift that probability mass that is sigma i star of si to si prime. And because and then you take the expected utility and the because of the fact that this is strictly larger, you can show that the expected utility that you get in the so called mixed strategy Nash equilibrium is going to be smaller than the expected utility that we are just now constructing. And therefore, that is a contradiction to the case that it is a mixed strategy Nash equilibrium, you already found a competing strategy profile which gives you a larger utility than the mixed strategy Nash equilibrium. So, therefore, it is not possible. So, what we assume for contradiction is not true. So, definitely it must be the case that this inequality should be greater than or equal to. So, that proves the condition number two. Now, we have proved the forward direction that if it is an MSNE, the condition one and condition two are getting satisfied. Now, we will have to prove the other direction that is if we are assuming that these two conditions are true, condition one and condition two are true, then it must be a mixed strategy Nash equilibrium. So, how can we prove that? So, just reduce the notation a little bit because we know that condition one is saying that for all the strategies in the delta sigma i star, the support of sigma i star, it is going to be same. So, we can just represent that by a number which only depends on sigma minus i star. So, let us call that Mi of sigma minus i star. And also what we know is that that is going to be at least as large as any other strategy outside that and also inside. Inside it is equal, outside it is at least as much as the utility of those strategies. So, we can without loss of generality say that this it is the maximum over all the strategies in the strategy space of player i. So, this comes from the condition two. So, these two things are essentially rewriting the same conditions. Now, let us look at the utility of sigma i star sigma minus i star. As before we are writing this as a probabilistically weighted sum of the utilities as at those pure strategies where the pure strategies are living within the support of sigma i star. And this is by the definition of sigma i star for all other cases it is going to be zero. So, we do not really need to care about that. Now, we know that because these numbers are all equal. So, we are looking at only the strategies which is within the support. So, these numbers these infinities are going to be the same which is equal to Mi of sigma minus i star. So, this is a constant it is not it is not dependent on Si anymore. So, we can take that constant term Mi of sigma minus i star outside the summation and the summation will always be equal to 1. So, therefore, it just becomes Mi of sigma minus i star. Now, we use the condition two. So, we are just looking at the maximum value of Ui of Si sigma minus i star. And using the previous observation we know that this when you are trying to maximize something over. So, the maximum value that you get when you are maximizing over this individual strategies or individual numbers that is equivalent to maximizing with respect to a mixed strategy, maximizing with respect to a convex combination of the same things. And that is definitely going to give you this expression here because this is the maximum value of over all the sigma i's. This is going to be larger than if you pick any specific sigma i and this should hold for all sigma i in the mixed strategy space of player i. And that is essentially what we wanted to prove. So, we wanted to prove that if you look at sigma i star sigma minus i star and player i makes a unilateral deviation to sigma i for all such sigma i's, this inequality is going to hold. And we have assumed an arbitrary player i and therefore, this inequality should hold for all i in N. And therefore, we have proved that this particular strategy profile sigma i star sigma minus i star is a mixed strategy Nash equilibrium. So, that proves both the directions of this characterization theorem.