 Let us just do that as an example also ok. So let us take this case 1, 3, 0, 6, 2, 7 ok. So now here what is happened is you have the column player has 3 pure strategies. So if I want if so earlier you had just 2 pure strategies. So you had z1, z2 you could write z2 in terms of z1 or z1 in terms of z2 and you could plot with respect to one of them right and so you could plot a unidimensional plot. Now you have z1, z2, z3 and if you want to plot even if you eliminate 1 you will need you will still have 2 variables for the other way. So you will need a 2 dimensional plot ok. So there is some still some trick you could use in some cases ok. So let us try to let us try to see how that can how that can be used here. So let us first first we will start with the with the row player because he has 2 strategies ok. The row player has 2 strategies. So what we will plot? So start with y2 equal to 0, y2 equal to 1, y1 equal to 1, y1 equal to 0 ok and let us start let us plot max over j, y transpose aj ok. So let us take j equal to 1 first. So that means column 1. So column I need to mark this. This is starting from say 0 here 1 for column 1 that means you have so you start from y2 equal to 0. So this is y1, this is y2 ok. So y2 equal to 0 which means that you are starting at value 1 and increasing to 6 right ok. So from 1 to 6, j equal to 1 ok. Now let us take let us take j equal to 2, j equal to 2 this will start from again y1 equal to 1 which means 3 and decreasing to 2. So you start from here and end somewhere here ok. And the last guy is going to start from 0 and end at 7. This is for j equal to 3. Now what is the max over j? What is the upper envelope? It is going to be this, this and this right. This is the max over j. So the red region here is the max over j of y transpose aj ok. And now what is the min over y of this? Min over y of this is this one ok. So and can someone tell me what is the what are the values here? So it is the it will have j equal to 1 and it is the intersection of j equal the j equal to 1 line and the j equal to 2 line right. So j equal to 1 line and j equal to 2 line at least that as per my drawing. So can you calculate? Yeah it is 1 by 3, 2 by 3 right. So sorry 2 by 3 I think for y1 right if I am y1 equal to 2 by 3 ok. So you should get so this is y star which is y1 equal to 2 by 3 and y2 equal to 1 by 3. Yeah and 1 yes. So we got our security strategy for the row players. So that is equal to 2 third and 1 third. Now what do we do for the column player? As I said the column player you would now have z1, z2, z3 here right. You would have z1, z2, z3 for the column player. Now what do we do about him? Because but how do we use the height ok and but that to remember that is like saying you are just looking for a best response. There will be many such z's that will give you the same value of y transpose, y star transpose. There is something in this picture actually which you can use. So see what are we plotting here? We are plotting here the payoff that the column player would get or any player would get ok as the column player changes its pure strategy. So if the column player was playing j equal to 1 is pure strategy which is the first column this was this line, the line was this one right. If the column player shifted to 2 then it became this line. If the column player shifted to 3 it became this line right. Now the column player is trying to maximize ok. So if he plays 3 in response to y star what would he get? What is the value he would get? If he plays 3 to in response to y star he would get this smaller height sub this height from here till here right. So he would get basically if he plays 3 he would get just this height clear. Now if he plays 3 in combination with let us say in combination with 2 ok 3 in combination with 2 he would get what would he get in response to y star. So if he plays 3 he gets this height if he plays 2 what would he get? He would get this this whole height right. If he plays 1 he would get again this whole height which means what? He should not play 3 at all right which means 3 is actually giving even a slight positive weight to 3 is suboptimal for him. So it is so the row player by playing y star has made the column player indifferent between 1 and 2 and now the column player has to choose between 1 and 2 he has to choose a do a randomization between 1 and 2. So in other words what you have to do is effectively although z3 is there in the problem you can take z3 to be 0 in the because that is the nature of the best response it will always put z3 equal to 0 and then you can draw basically the corresponding one between for z1 and z2. Now you have just two strategies for the for the column player and proceed as before and you will get you will get your z1 star you will so you will have to plot here again let us just do it for completeness sake. So again z2 equal to 0 here in z1 z2 equal to 1 this is z1 equal to 1 z1 equal to 0 0 let us plot this so can you tell me what is A z i for each i. So I am taking y so take i equal to 1 and z3 has been taken as 0 here as a function of so tell me what what would it be so for i equal to 1 which means you are looking at this row here and since z3 is 0 this is this is this is to be ignored so it is you have only these two. So you start from 1 and end at 3. So you start from 1 and end at 3 the other case you start at 6 and end at 2 and again your lower envelope is now this one and what you want to your z star is this and someone calculate and confirm. So what is z1 is what z1 is 1 by 6 and z2 is 5 by 6 can you check if this is this gives you the same height here so this value here is 16 this height is 16 by 6 okay something between 2 and 3 I can see that 16 by 6 and that is the value here. Now you can do this whenever one player has two strategies the other player may have any number of strategies what will happen is generically you will have since this is coming to see so what happens here is if you see y star y star is generically will be formed by the intersection of two lines there will be a few coincidences when there will be multiple lines passing through that same point but y star will usually get defined by two lines they you know could have a case where there is a third line passing exactly through this through this in which case you are you are kind of stuck but otherwise it if it is formed by the intersection of two lines all other strategies except for those two are being are basically suboptimal for the player and they are to be eliminated not necessarily may not be may not be a unique security strategy here. So if all three pass through the same point then you have to basically explore the whole the three dimensional space. So in fact the lack of a unique security strategy could be for other reasons also see for instance it could happen that you know the there is a there could happen the shape of this could be that there is a plateau somewhere in which case there could be multiple security strategies that could very well happen. So anyway this is in two dimensions it is these things these kind of things would not happen that easily and except in you know very contrived cases but the point is that generically what you will have is that this point is defined by the intersection of two of these lines so you limit yourself to only those. Now that just as a point of caution that does not mean that so just the way we eliminated j equal to 3 here we eliminated it because it it gave a lower value than these these two for y star for the when the row player played y star that does not mean that j equal to 3 was dominated. So in particular if you see this part here j equal to 3 is actually better. So there is some strategy where j equal to 3 is better. So it could not have been eliminated through a you know some kind of a domination argument. It has it is it is just not a best response to y star that is it and that is why it is being eliminated. So it is not being eliminated in the in the in the sense that it is not being removed from the game we are just eliminating it in our calculation in finding the z star. Two equations in three variables yes as far as if you if the only equation you are trying to solve is so if you wrote out this so you wrote y star transpose A z equal to now let us call this height h equal to h you will have three multiple variables here solving this but that is not what we are solving for we want to do the max min we will have to do max min over z see the point is you will still have all three z1 z2 z3 or at least two of them which is z1 and z2 when you are doing min over z of see how did we find this part this one here we did max over z of min over y. So when we did this part that is what we were doing. So here in doing this we eliminated z3 now you if there are three lines passing through the same point all you will have both you know z1 and z2 you will have to plot this in a three dimensional sort of a plot like this here z2 z1 then try to plot something etc. So that would be the only only the see in general what is this as I told you once you go to higher number of strategies and dimensions the min max theorem is basically linear programming duality so it is as good as having to solve a linear program. So if you you do not expect it to be much easier than that in general. Now can someone tell me what would a picture with I just said that j3 is not to be j3 is not dominated here but if it was dominated how would the picture look yeah right. So if j j equal to 3 was dominated the picture would look like something like this that you would have you have your j equal to 2 you have j equal to 1 and now j equal to 3 was suppose dominated by j equal to 1 this is j equal to 1 j equal to 2 if j equal to 3 was dominated by j equal to 1 it could be something like this which means regardless of what the row player plays j equal to 3 is worse for the column player than playing j equal to 1. So that means it is always below this does not have to be parallel or anything just below that is it. If it is dominated by both 1 and 2 then in that case in that case it would be something like this j equal to 3 it could be dominated by a mix combination also that is also possible. So in fact I have not talked of domination in mix strategies so it could be dominated by a mix combination that is possible not necessary because it is uniformly it has to be for every strategy of the other way. So that is a little more so domination in mix 2 will have to be you will have to try you have to write that out carefully. And now the final point yeah we have all the other properties that we said you we have for saddle points in pure strategies we have now that there is a saddle point we know there is always a saddle point in mixed strategies we always we have these properties such as for example that so every saddle point has the same value we have that if y1 z1 and y2 z2 are saddle points then so are y1 z2 and y2 z1 so this interchangeability property also. And like I said before that every saddle point okay so all of these properties that we used so we actually use the last one that you know to find a saddle point and the value it suffices to just find security strategies and that is how we just we just went about finding the min max and the max min by drawing those two big figures. So now as we move to non-zero sum games here is something that here is a counterintuitive example of the kind of things that could happen in a non-zero sum game. Now the zero sum game because the reason everything could be reduced to saddle points or security type of thinking was because we the damage anything that the other player does what damage to you right there was just no way for gain for one player to be not gain for the to be also gain for the other player okay. Now as a result of this you know this kind of analysis becomes very easy now here is an example of a non-zero sum game and let us see I will just show you what happens in this okay. So suppose so you have two players and two strategies up down for the for player one player one up this is for player two and left right for player two the let us take both players are maximizing okay both players maximizing the payoffs are like this so payoffs as 1, 3, 4, 1, 0, 2 and 3, 4 okay. Now what is the solution of this is there anything is there any dominance here they both are maximizing so U dominates U dominates D right U dominates D for player one okay so U dominates D so so and in fact also you can then after that you can eliminate so you can eliminate D and then eliminate R and you find that U L is the U L is the only Nash Equilibrium okay so U L is the only Nash Equilibrium and in once U L is the Nash Equilibrium player one gets player one gets what he gets one player one gets one and player two gets three in this okay. Now suppose let us take a hypothetical situation suppose player one decides that I am going to do damage to myself okay so player one actually reduces his payoff from his Equilibrium strategy okay the dominated the dominant Equilibrium strategy he reduces his payoff and reduces it by say 2 okay so now the payoff becomes the new payoff matrix now becomes this so player one goes and does damage to himself by two units whenever he is going to do he is going to play U okay so he one becomes minus one four becomes two the other things remain the same now what happens so now now D dominates U and L after elimination L is dominated by R okay and what you are left with now D R becomes the Equilibrium okay and in this case by you know by doing this kind of self-inflicted harm to himself in U what has he got effectively in the new game player one has now got player one has now got three and player two has got four player one has actually become better off so this this is not a model as a part of a strategy okay to do this kind of harm but this essentially what I am what is this pointing out this is pointing out that the outcome of a non-zero sum game can change very dramatically when you change certain certain numbers so a player can actually his payoff pump his winning or dominant Nash equilibrium strategy if it reduces in the new game the resulting game he might actually end up being better off okay so it can happen that you know what is on the face of it seems like damage to you might actually end up being better for you in the in the in the resulting new game okay so this is the reason for this is is the non-zero sum nature of this of the non-zero sum nature of this problem essentially okay because yeah yeah this is okay what do you mean yeah no no no that is okay but but you mean you which cost do you mean the second one the first one no it is it is a but dr is but he for for the row player ud u is playing ud is u is better than d no no he is maximizing they are maximizing no no maximizing no so u is better 4 is better than 3 1 is better than 0 no no this is yeah it is not just about the exchange of influence the point is there is a non-linear very highly dramatic changes and highly non-linear changes that happen when you change when you change you know in to an to the outcome of a game like this okay so what might seem like like damage on the face of it can actually end up being better for you or better for the player at the end in fact better for both players also in this case okay