 So far we have defined a bunch of different notions starting from dominance to Nash equilibrium and finally we have also talked about the max mean So where dominance we have seen that it is it cannot explain all reasonable outcomes because there are games where no dominant strategy exists Then we have seen pure strategy Nash equilibrium where unilateral deviation is not Beneficial which gives some sort of a stability guarantee and max mean is essentially a rationality for the risk aversion when you cannot when one player is not sure about whether the other player is going to play according to the to the rationality and intelligence That we have assumed then it's gives some sort of a security that even if they play that max mean strategy They are guaranteed to get that max mean Now what we are going to see and we have also seen a technique of finding the pure strategy Nash equilibrium via iterated elimination of dominated strategies now In this module we are going to ask what happens to these notions like pure strategy Nash equilibrium and max mean properties When you eliminate the strategies eliminate some of these dominated strategies So we have seen this example earlier that if we if we remove some of these strategies Some of the weakly dominated strategies based on in which order we are removing this Strategies it might end up in two different reduced games and therefore the the predictions could be different But the question is does it change the max mean value? So let us look at the one example for player one So let's say We are first looking at player one and we find that B is a weakly dominated strategy for player one Because M weakly dominates it. We have seen this in the earlier module So if I remove this what happens so before removing the strategy B What was the max mean value for player one? So let us just it is this part. So what was the max mean value for player one? So in for this row the maxima The minima was zero Here the minima was two and here also it was zero. So the max mean value was two for player one Similarly, if you look at the columns, this is the minimum value here This is also the minimum value here zero and this is the minimum value for the player two So the for player two the the minimum value the maximum of the minimum value was zero So the max mean value was zero now once we have removed this dominated strategy B then You can see that the minimum value for player one Is two here and minimum value zero here. So the max mean value for player one remains two while for player one Sorry player two the Minimum value here is two here the minimum value is one and here the minimum value is two again So the max mean value will not be one. It will be two Okay, so we can see that if I remove the strategy So for the player whom I am removing the strategy that players max mean value did not change But it might have changed the max mean value for the other player in this case player two So this is the observation max mean value is not affected for the player whose dominated strategy is removed And this is not just a coincidence In fact, we can formally prove that this is this is what is going to happen for the player whose dominated strategy you are removing Her max mean value is not going to get changed So let us formally state that in the form of a theorem So consider a normal form game, and we know how to represent the normal form games the the strategy the The player set the strategy set and the utilities of all these players and suppose we identify one strategy SJ hat which is a dominated strategy for player J and Once we remove that the residual game is denoted by G hat After removing this strategy the dominated strategy is J hat So the theorem claims that the max mean value of J in this residual game is equal to the max mean value in the original game G So before jumping into the the formal proof, let me give you a an intuition Why this is going to happen? So if you look at the the previous example, you can see that when we are trying to Eliminate when we were eliminating the dominated strategy for player one What is going to happen? the max mean value is the maximum of all the minimums and The only way you can change the max mean value is if you remove that strategy Now we are only removing the strategies which are dominated and you can you can build the intuition that this The that strategy cannot be a dominated strategy because it's a max. It cannot be a dominated strategy So that's the that's the intuition here and we are going to make that a little more formal So let us start with the maximum value of the of this player J in the original game G so which is denoted by VJ lower bar and By definition, it is the maximum over all SJ's Which is in capital SJ and minimum over all is Minus J's and this is the this is the definition of that utility Definition of that maximum value now. Let us look at the residual game the G hat game Where we have It's the same expression the only difference that has happened is now the strategy is set For player J has reduced by one and it does not have that SJ hat anymore Fair enough now because SJ hat is a dominated strategy in the original game Then there must exist some strategy some strategy for the same player which is certainly in the in this set in the set SJ minus this SJ hat that means this This strategy also lives in the residual game Which is at least as much as the utility Of that player when he plays SJ hat and this is going to be true for all is minus J's No matter whether this is strictly or weakly dominated This inequality because this is the weaker inequality. This is going to hold for all strategies for all is minus J's Good, so we we know these two results now we are going to make a sequence of Implications of this two observations So first thing is that if I just do Do a take a minimum over the same thing? So this thing here on the left hand side if I just take the minimum So and suppose that the minimizing strategy profile of all the other players is denoted by S minus J Hat so this is this is just denoting the same Minimum value of that minimum value of that utility of player J when player J is picking this strategy TJ and then we know that because this is This strategy dominates the strategy S minus I SJ hat So therefore it it this inequality must be true And this is a direct consequence of this it should hold for all S minus J's So in particular for this S minus J tilde also this inequality should hold so this is This inequality is obvious Now this last inequality is coming because of the definition of a minima So again, we are using the same old trick that If you pick one specific S minus I J tilde Then it must be at least as much as the minimum over all my S minus J's And when the player is playing this S minus SJ hat Okay, so Putting these two things together. We know that this part this minima is greater than this minima here right Now what we are going going to do on the on the this part So we are just going to take the maxima over all the All the strategies in the residual game So we are looking at the strategies in the residual game We are maximizing over that Minimum value. So this is the this is the minimum value And that will definitely be at least as much as the left hand side of this quantity left left hand side of this inequality Because this is a very specific strategy and because we are taking the maximum for all strategies In that set in the residual set Then this must be this inequality must be true T g is just one member of that set Now by the previous implication here in the previous line. We we know that this inequality also holds So what we have finally Said here is that this term Which we know is the V V lower bar. So we have also defined this so V lower bar j hat So this is the max mean value of the residual game. So V lower bar j hat Which is the max mean value If you want to compare that you can see that the left hand side is exactly the same as this So the max mean value of the residual game g hat Is something that we have shown to be at least as much as the minimum of Of this quantity when that player is playing S j hat and all the other players are so We are taking the minima over all the other players strategies Okay, so let's Mark this as one we'll use this inequality so we can remove this intermediate part We just want to remember that this inequality holds v j hat Lower bar is essentially at least as much as this quantity on the right hand side So how can we Make use of this thing. So let us start with the max mean value of the original graph. So this is v j lower bar By the definition, we know that this is max over s j in capital s j and mean over s minus j Which is in capital s minus j over all the utilities of that player Now we can divide this set s j is a is a large set Um, and there is one specific element here s j hat Which is a which is a dominated strategy. So we can actually partition this set all these elements which are On this side is s j minus this particular strategy is s j hat so The because we are taking the max it is a max over this as well as that So we can clearly write that this is a maximum over the same thing We have the same utilities here. So we are we are just copying it here And we are taking the first the maxima over this residual set And then taking the maxima over over the other So there is no maxima because there is only one element here So we can just plug that value in s j hat And if we look at this two values The maximum will be exactly equal to the maximum value that we have defined earlier Now what we already know from the previous example. So this this is nothing but so this part if you if you remember again This is nothing but v j lower bar hat because this is the maximum value in the residual game And we have already shown that that value is at least as much as the right hand side. So This value here is at least as much as this value which we have written here So the max value will certainly be the v j hat. There is no doubt about it. So this This particular part the first entry in this maxima is going to emerge as the maxima value of the whole Whole quantity which is the v j v j hat lower bar, which is the maximum value of the residual game So we have actually proved that the maximum value in the original game is the maximum value of the reduced game And if you look at this The intuition that I was talking about is because you your Strategy which is which you are eliminating Is is a dominated strategy strictly or weekly That particular part will not play a role in this maxima. So that is exactly what we have formally proved That intuition we have formally proved in this case