 So, now suppose if I gave you this zero sum game, simple zero sum game just with two strategies for each player. So, suppose Z1 and Z2 these are the components of the column players mix strategy, Y1 and Y2 are the components of the row players mix strategy. So, now what I will do is I will plot this as if so remember let us look at this from the point of view of the row player first remember further from the point of view of the row player I want to find a Y that minimizes the maximum over Z of Y transpose AZ and that is actually equal to the minimum over Y of max over j Y transpose AJ. So, we want to look at this particular thing we want to just we are just looking at this minimization over Y of the max over j of Y transpose AJ. So, what we will do is let us plot out for each value of j Y transpose AJ. So, let us take for example for j equal to 1. So, for j equal to 1 that means I am looking at column 1 I want to plot Y transpose AJ as a function of Y. So, keeping this column I want to plot this as a function of it. So, what is Y transpose A for j equal to 1? So, it is 3 Y 1 plus minus 2 Y 2 keeping j equal to 1. So, this expression here is equal to so for j so let me write this for j equal to 1. So, let us write out this expression this expression for j equal to 1 is 3 Y 1 minus Y 2 and for j equal to 2 what is it? It is 0 Y 1 plus Y 2. Now, we want to since there are 2 parameters here Y 1 and Y 2 so we will need to 2 axis to plot this but the simpler thing to do is let us we know also that Y 1 and Y 2 they together sum to 1. So, let us just plot them plot 1 at a time so we can just take it like this. So, let us take this point here let us plot everything as a function of let us say Y 2. So, this is Y 2 equal to 0 this point this is Y 2 equal to 1 equivalently this is Y 1 equal to 1 and Y 1 equal to 0. So, I am going to now substitute Y 1 equal to 1 minus Y 2. So, now can you tell me how does this look? So, at Y 1 equal to 1 or Y 2 equal to in short Y 2 equal to 0 you are starting from 3 and at Y 2 equal to 1 you end up at minus 1. So, let me just mark these out 3 2 1 0 0 minus 1 ok so this is the line for j equal to 1. So, I have plotted here I have plotted Y transpose A 1 ok similarly for j equal to 2 now j equal to 2 the line will start from where? From 0 and go to 1 right so it starts from 0 and ends up at 1 is fine ok. So, now tell me what is the max of max over j of Y transpose A what is this yeah going to be the upper envelope here right just this portion is max over j of Y transpose A j ok. And now what is the Y that minimizes max over j of Y transpose A j it would be the one the it would be the place where this red line has its minimum ok. So, that is actually this that is exactly this one here. So, this is your this is your Y star Y star and what is it equal to can someone calculate and tell me yeah. So, it is 2 by 5 Y 1 equal to 2 by 5 and Y 2 equal to 3 by 5 let us do the same thing here now for the column player. So, what by the way what does this tell us now? What is this Y star? It is a security strategy and the part of the and the component of the saddle point corresponding to the row player ok all right. So, let us now do the same thing for the column player ok for the column player again I will plot this with respect to z. So, this is let us say it is convenient to take I think z 1 equal to 0 here z 1 equal to 1 in other words z 2 equal to 1 and z 2 equal to 0 ok. And now can you tell me what the again let me mark these out here so let us say this as 3 2 1 0 minus 1 ok. So, tell me for let us say for the ith row that means for I equal to 1 the first row let us say for the first row what is what is going to be the so I should write out this expression here. So, now what we want to do is here we want to do max over z in z min over I A z I ok. So, let us take I equal to 1 and let us plot A z 1 here as a function of z 1 as a function of z 1. So, what are we looking at then we are looking at this this row I equal to 1 means this row. So, it is z 1 equal to 0 it is going to be at 3 z 1 equal to 1 it is going to be at 0 ok. So, you have from 3 to 0 then z 1 equal to 1 then for let us take for I equal to 2 this is I equal to 1. So, in short this is A z 1 this was j equal to 2 I transpose A. Now, let us do I equal to 2 now so what is A z 2 it starts from minus 1 and ends at 1 right. So, you have something like this ok so this is I equal to 2 in short. And now what is the max over z sorry ok what is the min over I of A z I the lower envelope now. So, that is going to be this one and f and now what is the max over z the maximizing z that will be this ok someone tell me what is it z 1 equal to 1 by 5 and z 2 equal to is this correct or is it the other way around. Yeah I plotted everything with respect to z 2. So, it should have started from 0 to 3 you know so take this as z so this is z 2 equal to 0 this is z 2 equal to 1 this is z 1 equal to 1 and this did not look like 1 by 5 to 5 in spite of my good drawing so ok thick. So, now here is the main point now ok so can someone tell me what do you think what should be these 2 heights what are these height they should be equal and why are they equal because they are actually because of the min max theorem right because max they are max min and min max of the same thing ok. So, these 2 are actually equal and they tell you the value of the game is this clear. Because this is the this here is min max and this here is the max min you can actually do this in any number of dimensions but if you want to do it manually obviously you need you know if you need basically 2 pure strategies per player ok because otherwise you will not be able to do it now however you can in some cases work with just one player having 2 pure strategies ok. So,