 So, the reason I got we got into common knowledge was because we were talking about dominance and we were talking about and in that the common knowledge of rationality was used then I said let us get into a formal definition of common knowledge. The main point of behind telling you all this is a exposing you to this result firstly and secondly also sensitizing you to the fact that one has to be there is a formal and rigorous way of thinking about games and this is the sort of development that goes in the background in establishing any properties. If you want to derive any properties about games, this is how you need to go about developing it. So, now let us study specific game classes and the simplest class and also the earliest class that was studied is what is called as the class of zero sum games. So, I will consider the case where the there are finitely many strategies for each player. So, that many times the math is easier. So, finitely many strategies for each player. What is a zero sum game? A zero sum game basically comprises of two players. So, there are just two players player one and player two and the utilities of the players are such that they sum to zero. So, you have if you take utility of player one when player one plays strategy X1, player two plays strategy X2 that is equal to negative of utility of player two when he plays when player one plays strategy X1, player two plays strategy X2 and this is true for all X1 and X2. So, regardless of what they end up and the profile of strategy is chosen by the players, the payoffs that they get are exactly negative of each other, which means profit to one player is necessarily loss to the other player. There is in such a situation this is a very this sort of problem classes is instructive to study because in such a situation there is no scope for players to hope for cooperation. There is nothing for them to cooperate on anything that one gains is the law is a loss to the other player. So, therefore, there is no dilemma about cooperation and therefore, therefore, in many ways the analysis is also a lot easier. So, in you can say in this case payers are sort of enemies, you know in essentially whatever one loses is gain to the other. Now, we since we are going to consider these players with finitely many strategies, one way to write this out is is write out you now the is is just as a matrix. So, you can write you as a matrix, let us call this matrix A. So, you can arrange these matrix in this photo in in a following way, you have the rows of the matrix are so rows are the strategies of player 1 and the columns are strategies of player 2, they are strategies for player 2. Now, what do I mean by arranging this is the matrix, I you can arrange this in the matrix in the following following way, essentially every entry here, now corresponds to just the entry in a in in in in any any row comma column of this matrix is just comprise is just depicting the payoff of that player of player 1. So, this here is the payoff of player 1 or utility of player 1, this is simply denoting you want. And the reason I am denoting only you want here, so you want this is just you want of i comma j. So, this is ith row and jth column, the reason I am denoting this by u 1 of i comma j is because is simply because u 2 is going to be just negative of this. So, it is enough to denote everything with just one number which and that is depicted in this matrix. So, in other words what we can think of is the following that player 1 is who is picking the rows and player 1 is looking for the least value in this matrix, player 1 looking for the least value in this matrix because he is looking to minimize u 1 or whatever cost. So, we are always thinking of minimizing. So, player 1 is looking for the least value in a by by choosing rows and player 2 who is the one who is choosing columns is looking for the largest. So, in other words the entire game zero sum game can be depicted with basically just one matrix. One matrix and the what is going on is player 1 is looking the row player is looking to choose a row to get the least possible value, the column player is looking to choose a column to get the largest possible value. Now, given this given that this is what players are looking to do. So, let me just denote this thing here i a ij is now the cost or whatever your player 1 when player 1 chooses row i and column and player 2 chooses column j, all right. Now, given that this is what players are trying to do the each player is trying to get you know, pull in opposite directions basically player 1 wants to get the least value player 2 wants to get the largest value. One possible way of playing is is that each player looks at the worst possible damage that the other could do you because anything that the other player does in his own benefit is damaged to me right. So, then I can ask what is the worst damage that this guy the other guy could do. So, what how do I quantify that well one way of talk the way to quantify that is the following. So, I can look for so player the row player can look for a row i star such that so what is the maximum damage that he can get when he plays a row i suppose player suppose the row player plays row i what is the maximum damage that he can suffer he wants the least value of a ij right so what is the maximum damage that he can maximum over j right because the largest value the worst case that could happen is that the j could be is chosen such that this is the largest right the maximum over j of a ij and now the row player can say well I can choose an i such that the worst case damage is minimized I can choose my i such that the worst case damage is minimized that means this is my worst case damage this is the worst case damage from playing playing i playing row i right this is the worst case damage from playing row i that is equal that is this and you want to minimize this worst case damage which means you choose your i such that this value which is now a function of just i this value should be least so you choose a row i star such that this is this guy is greater than equal to you choose a row i star such that the worst case damage that that you get from i star okay is less than equal to the worst case damage that you can get from any other row remember he is looking for the least possible value but least possible value in what it depends also on what the other guy would do so he is saying okay let me take the worst case of what the other guy could do okay so here he is taking the worst case of what the other guy could do when he when the row player chooses i star this is the worst case of what the other guy could do when he chooses i and he wants this to be better than the worst case in this. Now, this is how row, this is, so I can define a row i star in this way. So what is this essentially doing? This is basically a row i, a choice of row i star like this is securing a certain payoff for the player. It is basically telling him that well I can at least get this much. Because this is the worst case, this is the one that this is what this is the worst case. This is minimizing my worst case damage. Anything, if the player does not do the worst damage, I am still better. And any, if I do not, so or any the, any other, if I had chosen any other row, then I would be, you know, they would be a possibly a case in which I could be worse. So this is minimizing my worst case damage. So now, so what this does is, so this is ensuring basically it sort of curtailes his loss, says that well you cannot get, you know, worse than this. So this we can actually give it a name because it has some kind of strategic consideration. So we can give it a name and we can call it player, this, this here can be called the row players security level. So in other words it is and we denote this by v upper bar. Now and the i star itself is called the row players security strategy. So in other, let me write it in here v upper bar of a is simply this, it is the minimum over i of the maximum over j of a i j it is. So you look at the worst case over j and then you minimize that over i that is, that is v upper bar. So I have reason now from player 1, player 1's point of view, the player who is choosing the row. I can do a symmetric reasoning like this from player 2's point of view also. Player 2 would also want to look at the worst case damage and minimize the worst case damage for himself. So what would that give you? So you would then choose a column, you would choose a column j star such that, so if he chooses a column j what is the damage that he, worst case damage that he could incur? It is, it would be the minimum over i of a i j, right? The row player is going to choose the least possible value. So he looks at the minimum possible value of a i j over all i and he says well now let me try to get the largest such value by choosing the j. So you look for a j star such that minimum over i j star and this again is now, this is the security level of player 2 and j star is called the security strategy. The security level of the column player is denoted v and above of a, it is a function of just a. So v under bar of a is maximum over j of the minimum over i of a i j. Is that clear? There is necessarily a unique security level for each player because it is defined this way. There is at least one security strategy for each player because again of the way it is defined there could be multiple security strategies. So the security strategies for both players then provide us a way, a kind of a suggestion or a way of solving the game. We can say well each player plays, can be said that you, well anything that the other guy does is damage to you. So you look at the worst case damage and play accordingly, right? And then you look for and so essentially the pair of security strategies can be thought of as constituting a solution. Where would this logic fail? What is not equal? So as a concept where would this fail? So I could think of a solution concept in which both players play security strategies, may not achieve the security level, correct? So in short it is possible so that row player plays i star which is his security level. But the column player has a best response which is not j star or the other way around column player plays j star but the row player's best response is not i star. In that case there is room for each player to get better than the worst case that they were imagining, right? And then there you are basically at a situation where you can say well this is not a final resolution of the game because there is at least one player who would want to do better than what he is doing in this situation. And then the whole analysis changes, right? So let us think about this a little more carefully. First can you tell me what is the relation between v upper bar and v lower bar? So v upper bar and v lower bar are so v upper bar was what? v upper bar was min over i max over j aij and v lower bar is max over j min over i aij. So the v upper bar and v lower bar actually what they are doing is they are minimizing stroke maximizing but in a different order, right? In v upper bar you are first doing a maximization then followed by a minimization outside. In v lower bar you are first doing a minimization followed by a maximization outside. So how are these two quantities related? So actually the hint is in the notation itself. So v upper bar is, so you can very quickly argue that v upper bar is always greater than equal to v lower bar. So how do we prove this? This is a lemma. v upper bar is always greater than equal to v lower bar, why? Simple, the quick proof. So here is aij. Now I take the, so let us start with the left hand side just we are looking at v upper bar. So now this here is always aij. This is always less than equal to the maximum over j of aij unnecessarily and it is also greater than equal to the minimum over i of aij. And this is true for all ij. If you are getting confused with this, let us, I can put a i dash here and a j dash here. So aij is always greater than equal to the minimum over i dash of ai dash j and less than equal to the maximum of aij dash of aij dash, maximum over j dash of aij dash. So therefore, now what have I got here? If you see the right hand side here, the right hand side is a function of only j. The left hand side is a function of only i. And so you have a function of i which is always greater than equal to a function of j. So these are functions of two different variables. One is a function of i, the other is a function of j and the function of i is always greater than equal to the function of j. So it is always flying above the function of j. So its least value will be, the least value of the function of i will necessarily be greater than equal to the maximum value of the function of j. So which means therefore that min over i max over j, if you want I will still keep the dash here ij dash is greater than equal to, so I am just writing the extreme 2 and that is sufficient for us. So this is max over j, min over i dash, that is nothing but v upper bar greater than equal to v lower bar. So now what would be wonderful is if these two values would be equal because then what it would do is basically give you that the best response to a security strategy is a security strategy. The response of the row player being, thinking of the worst case is that the column player actually plays the worst case. And likewise column player imagining the worst from the row player in fact it ends up happening that the row player ends up playing the worst case. Unfortunately that is not the case and so let us do an example and actually we can see. So if these two were equal then basically it would mean that each player's projection about the worst case and each player's sort of way of this sort of this kind of reasoning where you think of the worst case that could happen and that ends up actually coinciding.