 If you looked at the example that we gave in the last module of the penalty shootout game and the mixed strategy profile that we have provided, you can find out that that is indeed a mixed strategy Nashic problem. In other words, you cannot really find any other sigma i prime or Si prime from which you can actually gain. Now the question is how did we find that MSME? That answer was somewhat like magical. So in this module I am going to tell you how actually that number, those numbers were found and for that we will have to develop a little bit of more definitions and notation. So first we are going to define the support of a mixed strategy. This is also the same thing as the support of a probability distribution. What that means is if you are looking at a specific mixed strategy which is nothing but a probability distribution of a specific player over the strategy space Si, then the support is the subset of that strategy space of that player on which this mixed strategy or this probability distribution has positive mass. So note that it has to be positive. We know that probability masses are always going to be non-negative but we are just isolating those strategies where the mass is positive. That is going to be called the support of sigma i. So whenever you are giving a specific probability distribution, you already know what is the support of it. So in that example, there was left and right. If the probability mass was positive in both these cases, then the support is both L and R. If it is 0 on one of them, then the other one remains in the support. So formally delta of sigma i is denoting the support of sigma i which is the collection of all those strategies in the strategy set where this sigma i has a positive probability mass on that strategy Si. So using this definition of the support, now we are in the right position to define the characterization theorem. So characterization as it means that it characterizes, so it gives some sort of a necessary and sufficient condition for the mixed strategy Nash equilibrium. That is what we mean by characterization results. Now what is this characterization result? We say that a mixed strategy profile sigma i star sigma minus i star is a mixed strategy Nash equilibrium. If and only if for every player i in this set of players n, the two conditions hold. First thing is if you look at the strategies in that, in the support of this equilibrium strategy for player i, so delta of sigma i star, I am looking at player i and I am looking at the support of that player i in this sigma i star. Then the expected utility of that player for all those strategies in that support when the other players are choosing the sigma minus i star is going to be the same. So this is the first condition that it has to be same for all the expected utility and remember that this is something that we were calculating when we gave that example. We are looking at what is the strategy profile of the other player and then we are looking at the expected utility of this player in one strategy versus the other strategy. They has to be same and in the example that we gave first 4 fifth and 1 fifth that was not true. And the second condition is that if we look at two strategies, so one strategy is living inside that support and the other strategy S i prime is living outside the support then the expected utility in that support must be at least as much as the utility of the other strategy which is outside the support. So this inequality should also be. You might wonder that I mean there is a more concise statement possible. So in some sense if you just replace this second condition with the S i prime to be inside capital S i then the second statement itself subsumes the first statement because then inside S i you also have this delta sigma i stars because delta sigma i star is nothing but a subset of S i. So for those cases this inequality will get satisfied on both directions. So therefore it should be equal. For the cases where it is not in delta sigma i star then this inequality should hold. But I have written this two statements separately just for the ease of understanding. This makes it more explicit in some sense. So what is the implication? So we will be proving this result in the next module but let us look at how this can be actually useful. So this particular characterization result will be very useful in finding out the mixed strategy Nash equilibrium in a normal form game. So let us go back to the penalty shootout game and see that there is no magic. So suppose so we will first have to find out the supports and there is no so this theorem does not tell you what that supports will be. It is just saying that for the support on which you have this mixed strategy Nash equilibrium those supports has to follow a certain property. The probability masses associated with that support has to satisfy certain conditions. But to find out the mixed strategy Nash equilibrium we will have to actually iterate over all possible support profiles. So let us look at the first support profile where both these players have L and L. So by this I mean that the support of the first player is L and the support of the second player is L. So that means the first individual the first player is putting all its mass on L and the second player is putting all its probability mass on L. Now this is definitely violating the second condition of this necessary and sufficient condition. Why? Because if you look at player one let us say what it is saying it is saying that if you look at the sigma minus i star so sigma minus i star in this case is just L. So you will be considering these two numbers minus one and one because r is outside the support and L is within the support and if you look at the utility so u1 when player one is choosing the action or the strategy inside that support which is L and the other player is choosing L this should be greater than or equal to u1 when you are picking the strategy which is outside the support and the other player is picking whichever it is picking. So this is the sigma minus i star but this is certainly not true this is minus one and this is this other part is plus one. So this inequality gets violated so therefore this cannot be a valid support profile. Similarly you can do this very similar exercise when it is r comma r you will see that again for these two numbers the inequality will get violated. So it is not even true for player for that support profile when both these players are choosing r comma r. Now the other possibility is that the first player and this is a symmetric case to the other case where player two is choosing the support profile L comma r. So here let us say player one is choosing the support profile of L and r together and the other player is choosing the support profile of L. Now this is not going to violate two at least in this case but what now violates is that the expected utility. So there is another condition the condition one which says that for all the strategies in the support the utility of this player must be equal to must be the same. Now again sigma minus one star in this case is just the L just one strategy and therefore in order to satisfy this condition one this number should be equal to this one which it is not so it is definitely violating the first condition. So even this cannot be a this case too cannot be a strategy profile it cannot be a valid support profile for a mixed strategy national equilibrium. And you can do the converse exercise when player one has L and the other player has L comma r and even if you do r and L comma r you will find the same outcome. Now the only possibility therefore is this case where both these players have the full support both the players has L comma r and L comma r. In that case we see that the condition two is vacuously satisfied because there is no strategy so let us go back here. So there is no such strategy which will satisfy this condition because the support is the whole of SI. So condition two is vacuously satisfied that is what we call there is nothing to verify in this case. All that we need to verify here is that this first condition is satisfied for both the players inside their support. So in order to do that what we can do is let us say we are going to put probability p and one minus p here and q and one minus q. So what we are going to do is that we are going to look at the strategies. So let us say for player one if we want to do the satisfy the condition one then it must be the case that the expected utility expected utility when it is picking this pure strategy L which is in the support and the other player is picking this mixed strategy sigma minus i star should be same when it is picking L versus it is picking R. So therefore you can then do this. So what it is doing is you are multiplying this minus one with the corresponding probability mass that this player is picking this L and this one with one minus q that is going to be the left hand side and similarly when you are going for R then it is one times q and minus one times one minus q that is going to be the right hand side. And you then solve this number you will solve this equation then you will find that q is equal to half and similarly you can do the very similar exercise let me just erase this part. So you do the very similar exercise by looking at the expected utility when you are multiplying this one with p and minus one with one minus p and equate that with minus one multiplied by p and one multiplied by one minus p when you do that you will find that p turns out to be going to half. So p and one minus p both are half and half so therefore the mixed strategy Nash equilibrium so this is the unique number which satisfies both the necessary and sufficient conditions of mixed strategy Nash equilibrium therefore this half and half for both these players is a mixed strategy Nash equilibrium and that is how we found the mixed strategy Nash equilibrium for this penalty shootout game and there was no magic there. So let me just for your own understanding of this idea how we were actually finding the mixed strategy Nash equilibrium let me give you two games and find the mix strategy Nash equilibrium as an exercise. So this games we have seen earlier so this is a game of football and cricket there were two friends one of them like football more than cricket and the other player like cricket more than football we have discussed that this games have pure strategy Nash equilibrium so two one and one two were pure strategy Nash equilibrium which will also emerge if you do this exercise step by step this strategy profile will also emerge as a MSNE or calculation please do that but there will be a non trivial mixed strategy Nash equilibrium as well so let me leave that as an exercise find there is a there is some p and one minus p and there is some q and one minus q but none of these are exactly equal to zero or one which what is that p and q and the second example is just a concatenation we are just you appending one strategy for the player two and the corresponding utilities try to find out what is the mixed strategy Nash equilibrium for this.