 The question is the problem unfortunately is that this is in general not the case, in general v upper bar will not be equal to v lower bar ok. So, if so as I was saying that if these two end up being equal then it would be that each player's projection about the worst case actually in short comes true right. So, we can formalize that by through this concept. So, suppose let i star j star be a player 1, player 2 such that a i star j star ok. So, you look at a i star j star with this is what is this quantity here, this quantity is the payoff that the players would get when they each play i star and j star ok. Let i star j star be a strategy such that a i star j star is less than equal to a i j star and also greater than equal to a i star j for all i and j ok. So, which what does this mean? That means that i star j star is such that when the if you look at it from the point of view of the row player here ok. So, there are two inequalities that i star j star satisfies, i star j star as this property that if you look at it from the point of view of the row player then assuming the column player is playing j star the row player responds with i star right is the best thing for him to do is to play i star and likewise assuming the row player plays i star it is the best thing for the column player to do is to respond with j star is clear. So, if you have these two such a if i star j star or if a pair of strategies is like this that satisfies this then i star j star are called a saddle point ok. So, this pair is called a saddle point. Now, I have used the notation i star j star for security strategies I am also using it again for saddle point and the reason for that is see if the if the security strategies together ended up forming a saddle point then you would have that then this then they would actually suffice to solve the game because effectively it would mean that each player when playing his security strategy the other guy responds also with the security strategy right because the response to i star is j star then the response to j star is i star ok. Now, how now this so if you could find such a pair i star j star that satisfies this is called a saddle point if you could find such a pair then this would actually end up solving our problem alright. So, we will actually let us actually prove that. So, theorem is this let A so, suppose let A be a matrix game such that V upper bar is equal to V lower bar then one then A has a saddle point ok and B second the saddle point comprises of security strategy. You can write out another part of this that if any pair of security strategies is a saddle point alright. So, so let us let us go about proving this. So, we already know that there always exist security strategies. Now, what we are going to show is that if V upper bar is equal to V lower bar and you put together that pair of security strategies they automatically form a saddle point ok. So, that is that will then show that A has a saddle point and secondly it will also show 3 which is that any pair of security strategies is a saddle point is this clear ok. So, that is what we are going to show ok. So, now remember we have we already know that V upper bar is always equal to V lower bar sorry what am I writing we know that V upper bar is greater than equal to V lower bar ok. So, which means that we know that max over j A i star j is greater than equal to min over i A i j star for security strategies i star and j star. But now what have we been given we have been given that these two are in fact equal ok. We always know that these are this is a greater than equal to but we have been given given V upper bar equals V lower bar that these are equal ok that is what we have been given. So, which means what therefore we have max over j A i star j is equal to the min over i A i j star ok. So, these are remember i star and j star are security strategies. Now what I need to show is basically that this actually i star j star in fact saw form a saddle point ok. So, how do I show that well remember min over i of this is always less than equal to A i j star and max over j of this is greater than equal to A i star j ok. So, now if I put and so this is true for all i comma this is true for all i comma j which means what now if I put i equal to i star and j equal to j equal to j star what would I get I would get A i would get equality throughout I would get A i star j star is equal to V upper bar A is equal to V lower bar A is equal to A i star j star alright. Now what this in particular means now remember what was V upper bar A V upper bar A itself was equal to the min over i max over j of A i j alright. So, in other words this was always always less than equal to max over j in lower bar A is max over j min over i A i j. So, therefore, this was always greater than equal to min over i for all j ok. But now what we what we showed was actually the these are each equal to A i star j star both of these are equal to A i star j star. So, therefore, now putting putting this is this is true for all i and for all j. So, I can write this for all i for in fact for i star and j star. So, what would I get I would get therefore, A i star j star putting these two together gives me. So, this is the minimum over i of A i j star. So, therefore, this is less than equal to A i j star for all i and also greater than equal to A i star j for all j. So, actually I do not need this to be less than equal to the point I actually need just this impact to be out here all I need is that this is impact the max over j A i star j and this is min over i of A i j star is this clear. So, I just showed that we upper bar we lower bar are both equal to A i star j star and because they are themselves equal to max over j A i star j and min over i A i j star that then gives me this input which means that which means that i star j star is a saddle point. So, what does what has this shown this is shown that both this as well as this. So, it is shown that A has a saddle point that is also shown that you take any pair of security strategies they will necessarily be they will they necessarily form a saddle point. You actually get one more thing for free here which I have which I have not specified, but that it tells you that actually when there is a saddle point right is V upper bar and V lower bar are in fact, functions of just the matrix and not the saddle and does not matter what the saddle point is whichever saddle point you choose if there is a saddle point whichever saddle point you choose the V the value is always going to be V upper bar and V lower bar is clear. So, so V upper bar and V lower bar were defined for from security strategies and therefore, they were functions of just a the saddle point was just defined abstractly as a bunch of inequalities. So, there is no guarantee that the value of that saddle point could be is has to be the same for every saddle point, but we are getting that for free whenever there is a saddle point, it will always be equal to it will always be equal to these this particular value. So, alright, so let us actually show the second part 2 here, which is that that a saddle point comprises of security strategies. Alright, so now what is the definition of a saddle point saddle point is that is the point I star J star such that I star J star if I star J star is a saddle point, then this must hold and what we will show is we will show that I star J star are security strategies. So, how do I argue this from this inequality I want to argue that I star J star are security strategies, which means I want to argue that I star is the one that minimizes the worst case damage for the row player and J star is the one that minimizes the worst case damage for the column player. So, how do I argue that we could try by contradiction, there is a simpler argument here. So, if you actually look at just this part, right, so this is the in fact the the interesting thing about zero sum game. See, if you look at these in isolation, any of these in isolation, look at this inequality in isolation or this inequality in isolation, this is only saying that I star is is the best reply to J star and this is saying that J star is the best reply to I star. It does not actually say that these two are, we are not really using the fact that these two hold simultaneously. This is just saying that each player is responding with his best reply, but that they end up giving the same value is not actually being used here. So, if you want to get back the security concept, right, then you have to use that you have to you cannot use it is not enough to use any of these in isolation. You have to use the entire chain of inequalities, which means that you have to use that this left extreme is actually less than equal to the right extreme. Now, what does that mean? That means that A I star J is less than equal to A I J star, okay? A I star J is less than equal to A I J star. So, but then what is A I J star that is in turn less than equal to A that is in turn less than equal to the max over J A I J, right? The right hand side is now, okay, I should maybe I will make it easier. I will first do this. So, this is the left hand side is a function of J. The right hand side does not have J. It has only J star. So, I can say well, therefore, max over J of A I star J is less than equal to A I J star, but A I J star is itself less than equal to max over J of A I J. And so, therefore, you get that max over J of A I star J is less than equal to max over J of A I J, which means that I star is a security strategy. So, this one here, I can observe that in this from this inequality on the first inequality here, left hand side is now not a function of I and the right hand side is a function of I. So, I can do a min over I over there, min over I J star and the left hand side will be, will still remain I star J, but then that is in turn greater than equal to min over I A I J. Now, combining these two tells me that J star is a security strategy. In other words, there is really, once the V upper bar and V lower bar are equal, there is no difference between saddle points and security strategies. Conversely, if a saddle point exists, then automatically you will get for free that the V upper bar and V lower bar are equal, because every the saddle point has to comprise of security strategy. So, if there is a saddle point, you will, they comprises of security strategies and you will get that, yeah, this saddle point value is actually equal to V upper bar and V lower bar.