 While explaining the matrix game we have actually given two examples and in one of those cases the psne did exist and the other case psne did not exist. Now have you noticed something interesting between these two games where psne exist and does not exist and the fact that max mean and min max were equal or unequal. Let me give you the hint. So in the first case the penalty shootout game we had this situation where the min max value was equal to one which was of course greater than or equal to and in this case it was strictly greater than the min max and the max min value which was minus one. Now these are not equal but in the other case the max min value and min max value both were equal to one and in that case psne existed. Is this just a coincidence? We'll show in this module that that is not a coincidence indeed if you have to have a psne if the psne has to exist then it better be the case that this min max and the max min value be the same. Now to prove that let us also develop a little bit of notation. So first thing is that we are defining the max min strategy of player one and this is just the strategy for player one which is arg max over this minima. So this is the formal definition looking at the minimizing strategy. So for a specific strategy s one what is the minimum value and then which strategy maximizes that that strategy we are going to call the max min strategy for player one. It's in some sense is the safest strategy or the worst case optimal strategy for player one and similarly because the we are using the same utility function which is negative the utility for player two is going to be the negative of that then the similar notion of max min for player two will be min max in this context for player two. So the min max strategy of player two is something that we are going to denote with s two star. So this is the min max strategy for player two. So here is the theorem which formally states which we have just said earlier. So a matrix game u has a psne or a saddle point if and only if this min max and the max min value are equal and this s one star and that equals the the value of this matrix at s one star and s two star where s one star and s two star are the max min and the min max strategies for players one and two respectively and not only that in particular this particular strategy profile is going to be the psne up psne of this game. So in order to prove this we'll because this is a necessary and sufficient condition so we'll do both directions one by one. So first look at the only if direction. So this is the only if that is we start with the psne and we show that this condition holds. So we will start that this game has a psne and we'll show that this this inequality is true. So suppose this psne is s one star s two star I mean don't really need to be the same s one star s two star it will happen to be the same but for simplicity we are using the same notation that we have used in the previous modules. So by the definition of psne what we know we know that this inequality holds for player one and we also know that the other inequality holds for player two of course because this is looking at you then in that case the inequality will get flipped but that will also hold for all s two. So a similar inequality we can write even for player two. So let us start with this one. So because this is the this is greater than that we can actually so this is this inequality is true for all s one in capital s one. So we can actually write this even for the maximum value of this this inequality is going to hold. In fact in that case it is going to be equal but let us use the same argument and we are using just this inequality and we are taking the maximum. All right because this is a this is maxima and because s two star is a specific strategy therefore we can clearly write using the definition of minima that we can say that this quantity which is which is a fixed quantity only dependent on s two star because it is no longer dependent on d one we have already taken the max. So this is going to be at least as much as the minimum. So minimum will always be less than or equal to that number so we can write this min max value here. So and if you write it in this way we know that this is the min max value of this matrix game. Okay so now we are using the same argument for player two so you can write an equivalent expression. So what will that be? It will be s one star s two star that is going to be less than or equal to u of s one star comma s two and that will hold for all s two in capital s two and then you can use the same thing but now the inequalities are flipped and you are going to use first the minimum value and then the max and therefore you will have the max min which is going to be larger than that. So this is the max min you can just redo the steps. The max min value will be at least at least as much as the utility of s one u of s one star s two star which is the same value here. Now what do we see? So we know that v upper bar is actually as at least as much as v lower bar but here we have already shown that this is greater than equal to the number s one star u of s one star s two star and it is also upper bounded by the same number u of s one s one star comma s two star. So this is these two numbers are actually sandwiched between between those two things. So this is by the previous lemma and this is what we have just shown. Therefore these two numbers must be equal to must be the same and they must be equal to this u of s one star comma s two star and that is exactly what we wanted to show. This s one star and s two star are going to be this yeah so this is if it is a saddle point then these two things are actually going to be the going to be the same and because this min max values and so the this s one stars and s two stars are nothing but the saddle points they are also going to be the the min max and the max min values. So yeah so the that implication of max min value is just coming because of this inequality because if you are if you have a utility at a specific s one star and s two star which is equal to the value the max min and the min max value then s one star s two star is both the min max and the max min max min strategy. So the we have shown the necessary condition that is the only of direction that if you have a psne which is s one star s two star then this inequality should hold and your s one star s two star which is which was psne must also be the min max strategy and the max min strategy respectively. Now we will have to show the other way around now we are given that this quality holds where s one star and s two star are the max min and the min max strategies for players one and two and let us for for our simplicity of notation let us denote both these equal values with v let us say. Now what we have now we know by definition of of minima again the same trick that we are going to use. So if you fix a specific s two and s one star is the same then this by the definition of minima you can always write this and because this s one star is a very specific it is a it is a max min strategy for player one then you can actually write it to be equal to the max of over this the strategy of player one and min over the strategy of player two. So s one star is by definition the the max min strategy so the equality should hold and because this is the max min value and we we know by by the what is given in this condition that is equal to the value v. Now similarly we can use the same thing so we we can use the same argument of u of s one comma s two star and we can use the the argument of max so because now we are going to use the other way other direction so we are going to show that this is max so if it is less than equal to the max with respect to the first quantity here and you use the same fact that because s two star is the min max strategy then you can use the same argument to show that this is min max and because that min max is also equal to v so the right hand side becomes v so you can show that this is also less than equal to v but now we also know that this v is equal to u of s one star comma s two star that is given in the very beginning. Now if you just write that down so what what did you get here s one star comma s two and notice that this is for any arbitrary s two this is going to be this whole implication that we have shown holds for any arbitrary s two that you pick so that is going to be greater than or equal to u of s one s one star comma s two star right so therefore the the definition of p s any is essentially this and from the second implication you can show that this is s two star and if the other player is is holding on to that strategy then this is going to be less than or equal to u of s one star comma s two star and this holds for all s two and this holds for all s one so if player one is deviating from this strategy to any arbitrary s one it is going to lose out and if this player player two is deviating from s two to s s two it is also going to lose out because its utility is just the negative of it so that essentially proves the theorem that we started with that if you have a p s any which is a saddle point it is going to be a saddle point if and only if this both this min max and the max min value become equal