 Now, let us look at a very special kind of a game or the matrix games. So, this is also known as the two percent zero sum games and which is spelling out the whole paradigm clearly there are only two players and their utility sum to zero. So, that is what a matrix game is. Why are we studying this kind of games? There are certain very interesting results some interesting properties that involves this stability notion that is the pure strategy Nash equilibrium notion that we have discussed and also the security notion that is the max mean strategies. So, we will see that there is a nice interplay between this max mean strategies and pure strategy Nash equilibrium in this kind of games. So, let us look into these games. So, as we know the games are generally represented by this couple of three entries, couple of three things. First is the player set in this case the player set is only two players one and two strategy sets and this can be arbitrary even for a matrix game and utilities. But utilities have a very specific structure that the if you look at the utilities and add them element wise the strategy profile wise then they will identically be equal to zero. So, what are the examples that we explain this notation a little better. So, I have just changed the previous penalty shootout game by removing one of the strategies for both these players. So, here the first player is the shooter and the second player is the goalkeeper. Now they have only two strategies either shoot on the left and shoot on the right or and for the goalkeeper dive on the left or dive on the right and the utilities remain the same. And what you can observe here is that the sum so the for every strategy profile l,l,r for every strategy profile if you take the utility sum of these two players they will always be equal to zero. Similarly, there can be an arbitrary game where both these players have three strategies each and you can see that every entry in this in this matrix has these two components which if you add it gives gives rise to zero. And because of this special structure you can represent this game in the form of one matrix. So, you can just ignore the second part I mean the in every entry there are two elements but you already know that the second element will be the negative of the first element. So, you can just represent the whole game just by using one single matrix. And let us represent that matrix by you or the utility of the first player the second player utility is just minus of you and that is the reason why it is called the matrix game you can represent the whole game using just a single matrix. Now let us look at some of the some of the things that we have discussed in the previous modules the max mean strategies and the min max strategies for this matrix. So, if you look at the max mean of of this matrix of the matrix u so in the first game that is the penalty shootout game the max mean is first finding the minimum over the rows so which is minus one and so that is that minimum value of that row is written here and similarly this minimum is also minus one so the max mean value in this case will be minus one in this matrix similarly when you compute the min max now you are taking the max with respect to the column so in this column you are taking the max which is one and similarly here also it is one so the mean of that two maximum values is also is going to be equal to one and these two things are not the same while for for the second example the second game if you compute the max mean value so these are the mean values for each of these rows and you find them the max of it which is going to be one and here the mean of this max values is going to be one in this case they are the same okay so now what is the relationship of your strategy Nash equilibrium with any of these things that we have discussed so far we are discussing different things max mean values min max values and what is the relationship of PSN is with all these things so first of all let us find out what are the PSN is of these games we know how to find out the PSN is let's go by the first principles if we try to find out the pure strategy Nash equilibrium of the penalty shootout game we see that that does not exist any this was one of our examples when we actually gave and gave a case for non-existence of pure strategy Nash equilibrium pure strategy Nash equilibrium is not guaranteed to exist because you cannot find any strategy profile so for instance in this strategy profile it is better for player 1 to deviate to R similarly if you look at this strategy profile it is better for player 2 to deviate to R here also because this is minus 1 so this is minus 1 comma 1 so here also for player player 1 it is beneficial for going beneficial to play L instead of R and here also it's better for player 2 to move to L so none of this strategy profiles for strategy profiles are essentially pure strategy Nash equilibrium alright so what about this in the other game here you can you can figure this out I mean take a bit of time and try to find out which one is the pure strategy Nash equilibrium let me give you the answer so you see that here in this strategy profile M comma R if you look at player 1 this particular number is the maxima of this row maxima over this column so this is greater than this and this is also larger than that similarly for the second entry it is also the row maxima so for player 2 R is is better than playing any of these other strategies CRL if the other player is playing player 1 is playing M and if this player is playing R second player is playing R then the best response for player 1 will be to play M so therefore M comma R is a pure strategy Nash now what can we observe how can we actually look at this look at this property so in in order to find out the the pure strategy Nash equilibrium in this kind of a game what are we actually doing we are actually trying to find out what is the the maximum value over this over this column and we are also trying to find out what is the maximum value over this column for the second entry now we know that the second entry is nothing but the negative value of of that number so if we look at only the matrix which is just listing out the the first entries here then what we are trying to find out is something which is which is on one dimension trying to find out the maximum value and the other dimension trying to find out the minimum value and mathematically those kind of points for for a matrix is what is known as a saddle point so here we are going to use the same name saddle point which is defined for the matrix but for for games so we'll call the this is the saddle point of this game and this the saddle point will refer to the value the the utility for this player 1 is going to be maximum and it is going to be minimum for player 2 which means that the in the other dimension it is going to be the the the minimum so if you want to represent the same matrix so this is the matrix representation of the same game and suppose there are multiple rows and columns and if we just try to portray this as so on this dimension I have x and on this dimension I have y so let us place this in a in a three-dimensional plane so I have this x which are representing the different actions or the different strategies of player one and y on the y dimension is the different strategies of player two then what we are actually trying to find out in a in a saddle point is something that that it will maximize when we are varying over x and something that will minimize when we are varying over y right that is the that is the this point if you can find any such point that is going to be the saddle point and we will show that that saddle point is nothing but a pure strategy Nash equilibrium in fact in the in the two person zero sum game or in a matrix game these two notions are one and the same saddle point means the pure strategy Nash equilibrium now you can you can actually try to do this exercise yourself so what are the saddle points of these two games you can see that in the in the first game the penalty shootout game there does not exist any saddle point because you cannot really find any such point which is a which is a row maxima or maxima over a over a specific column and it is a minima over the over a row so therefore no saddle point exists for that matrix but for the second example that arbitrary game example there exists a saddle point and we have seen that what is that saddle point so saying what we have just mentioned informally in a more formal way so we have this theorem in a matrix game with an infinity matrix u s1 star s2 star is a saddle point if and only if it's a psne so this is this is something which is actually making this notion of indistinguishability between this saddle point and psne so whenever we in in the context of matrix games whenever we are talking about saddle points we are actually meaning pure strategy Nash equilibrium and vice versa now what we are going to show this is fairly straightforward this this is a saddle point which means which implies and is implied by this is the notation for that it implies in both directions these two things so first thing is because it's a saddle point for the first entry here that is the when we are looking at the matrix and we are varying with respect to a specific column so this this way we are varying so then s1 star is at least going to be as much as any other utility in that matrix when the other player is actually sticking to that s2 star so suppose this is s2 star then s1 star is is a point s1 star is a point such that this the utility or the the value of that matrix gets maximized at this point now and this should hold for all s1s so therefore this is going to be a maxima so if you want to draw it in this way so this is the maxima on this dimension similarly it is going to be the minima when you are looking from the other dimension so if you are looking at s1 star so player one is now fixing its strategy to s1 star and you are varying in this way so you are going to s2 star and comparing that with different other strategies so for all such s2 star this point is going to be the minima so in other words this is if you want to draw it in this way so this is going to be the minimum point so therefore that is what the what a saddle point means but now we can see that by by the definition this is nothing but a definition of a pure strategy Nash equilibrium what you are doing is this is nothing but the utility itself is utility of one so the utility of one is nothing but u and utility of two is nothing but the negative of this u so therefore if you just flip this number so what you get is u2 s1 star s2 star and the inequality will get flipped u2 s1 star comma s2 so for all s2s this inequality is going to hold and therefore that is a psn right so that is the that is the that is by definition of the psn so so that that concludes this proof so we have so for that to formally establishes that psn and this saddle points are one and the same so now we are going to define the max mean and the min max values so something that we have already shown through examples so what is a max min value this is something that we have already done but now because we have only single matrix now we can we can just use the the the indices of that matrix u and we are trying to find out what is the minimum value with respect to the second coordinate and the maximum value with respect to the first coordinate and let us use this notation v lower bar to define that max min value similarly the min max value is denoted by v upper bar and it is just the it is first maximizing with respect to the first element the first entry here in the matrix and then trying to and then minimizing with respect to second element okay so now the question is how these two things are related and we have a small lemma which shows that for matrix games where this things this min max and max mean values are defined in the in the way we have defined it before this min max value is actually going to be at least as much as the max min value so this inequality should hold so note note that this is min max and this is max min okay so the proof is fairly straightforward you can just start writing this in the in a very standard way so first we start with the the the element element at s 1 comma s 2 and by the definition of minima you can find this this is the trick that we have used many times and now what you can find is this inequality is going to hold for all s ones right so this is going to hold for all s ones so therefore we can actually take maxima on on both sides so what we will have here is if we use the use the maxima on the on the left hand side and fix the strategy of of the other player to s 2 and use the maxima here so notice that on the right hand side it's just a function of s 1 it's no longer a function of t 2 because you have already taken the minima over it now after taking them the maxima over the first element this entity is neither a function of t 1 or t 2 it's it's a fixed constant now this inequality is going to be satisfied for all s 2 because we that is the only variable here now you can take the the minima so it this right hand side is just a constant number it does not change with s 2 now this is going to be satisfied for all s 2 and in particular if you take the minima then also it is going to get satisfied so that essentially completes our proof on the right hand on the left hand side you have the min max value which is nothing but v upper bar and on the right hand side you have the v lower bar which is the maximum value