 Welcome to this NPTEL course on game theory. In the last class, we have seen the zero sum games, matrix games in particular and we also have introduced the mixed strategies. Today, our goal is to establish this Minmax theorem. First we start with a general setup and then we specialize to finite games or matrix games. Let us start with the simple setup. So we consider 2 players, the player 1 strategy space is S1, player 2 strategy space is S2 and the payoff function is F from S1 cross S2 to R. So recall payoff to player 1 is Fxy and payoff to player 2 is minus Fxy, of course X is a strategy chosen by player 1, Y is a strategy chosen by player 2. Once again we recall equilibrium, saddle point equilibrium is defined as a pair X star, Y star, they belong to S1 cross S2 satisfying the following thing, Fxy star, Y star. When Y star is fixed by player 2, X star maximizes player 1's payoff and when X star is fixed by player 1, Y star minimizes the same function. This is true for all X in S1, Y in S2. As I said we need some assumptions, we assume S1 is compact and convex, similarly S2 is compact and convex, F is concave in X variable when Y is fixed, similarly F is convex in Y variable when X is fixed. So these are the assumptions required to prove our existence of a saddle point equilibrium. So let me state the theorem, let S1 and S2 be compact and convex subsets of some Euclidean space, F is concave convex function. What I mean by concave convex is that in the first variable X variable F is a concave function in Y variable F is convex, then saddle point equilibrium exists, this is the theorem. So we will try to prove this. The proof is not very hard but we require some play with the convex theorem. So first to start with we assume F dot Y as a function of X is strict concave and F as a function of Y variable is strict convex. So the advantage of this strict concave or strict convexity is that the function will admit a unique minimizer or maximizer according to its convex or strict convex or strict concave. So let us say what the strict convexity means is that Fx lambda Y plus 1 minus lambda Y prime is less than or equals to lambda Fxy plus 1 minus lambda Fxy prime, this is the convexity. In a strict convex case this is replaced by a strict inequality, so that is the strict convexity and a strict concaveity is defined analogously. So the first thing we would like to say is that the first fact under strict convexity the minimizer is unique, of course if I take concaveity then this minimizer becomes maximizer. But the proof is not hard, so let us say if Y, Y prime are two minimizers. This implies F, instead of writing F and all let me put it as a function G of lambda Y plus 1 minus lambda Y prime, this is going to be strictly less than lambda GY plus 1 minus lambda GY prime, this is strictly less than minimum of G, of course this is equals to this and therefore G of lambda Y plus 1 minus lambda Y prime is strictly less than minimum of G, which is a contradiction. Therefore Y has to be same as Y prime and hence uniqueness, the uniqueness automatically follows. Once we have uniqueness now let us start working out the details. Now for each X in S1, F X dot has a unique minimizer therefore let this be Y which depends on X, what it means is that F of X comma Y X is nothing but minimum Y in S2 of F XY, so let me I define this by MX. Now F is a uniformly continuous function on S1 cross S2, basically S1, S2 are compact sets and hence any continuous function on a compact set is uniformly continuous. So that is the reason why F is going to be a uniformly continuous function. Now this uniformly continuity implies that as a consequence of this is that MX is continuous function. So this follows from the uniform continuity, so the argument is that if X and X prime are sufficiently close, F X prime Y and F XY are also sufficiently close by irrespective of Y that comes from the uniform continuity and hence you can show that these are all, this MX is a continuous function I will leave this as an exercise, real analysis exercise. Next thing I would like to say is that MX is a concave function, Y is this. So recall MX is nothing but minimum over Y of F X comma Y, for each fixed Y, F XY is a concave function and then you are looking at the minimum of them, so minimum of the concave functions, so this is a concave function. This follows because MX is point wise minimum of concave functions therefore MX is a concave function. This implies MX has a maximizer, let this be X star in which is in S1. So X star is in S1, this existence of X star happens because S1 is a compact set and M is a continuous function on S1, so let this be X star. What we have is that M X star is nothing but max over X of MX which is same as max over X mean over Y of F XY. Now we will show that X star, Y X star is a saddle point equilibrium. So let us look at the following thing. First thing is that M X star which is same as F X star, Y X star. So recall Y X star is the minimizer of this function F, so therefore this is less than or equals to F X star Y for all Y in S2. So one inequality in some sense follows here, so this is one inequality that we want. When X star is fixed, Y X star minimizes, so now we need to look at the other way inequality. So let us look at that, F X star, Y X star, this is nothing but our MX star which is certainly bigger than MX for all X in S1. Now let Y tilde to be Y into 1 minus T X star not plus T X, basically it is not Y into it is a Y of 1 minus T X star plus T X for some T in 01, basically we are taking a nearby point around X star in the direction of X. So look at that, what is M of 1 minus T X star plus T X, this is nothing but F of 1 minus T X star plus T X star comma Y tilde and now F is concavity is there, so therefore this is nothing but 1 minus T F X star plus T F X Y tilde. So now what we have got here is the following thing, M X star is certainly bigger than M of 1 minus T X star plus T X, this is coming from the maximality of X star, this is greater than or equals to 1 minus T F X star Y tilde plus T F X Y tilde that is just proved that fact and this is greater than or equals to 1 minus T M X star, this is coming F X star Y tilde is bigger than or equals to M X star that is because of this fact, note that M X star is always less than or equals to F X star Y for all Y in S2, this immediately tells me that F X star Y tilde is certainly a bigger quantity than M X star, so that is exactly what we have used here, this plus T F X Y tilde. Now what we have here, let me write down here is that M X star greater than or equals to 1 minus T M X star plus T F X Y tilde, this immediately implies that M X star is greater than or equals to F X Y tilde, recall this Y tilde is nothing but 1 minus T X star plus T X of this is basically Y of the best response corresponding to 1 minus T X star T X and we have already seen that these are all continuous functions. Now let T goes to 0, this immediately implies M X star is going to be bigger than or equals to F of X as T goes to 0 this becomes 1, this becomes 0 therefore this is Y of X star. Now combining all these inequalities what we have is that F X Y X star this is less than or equals to M X star, M X star is nothing but F X star Y X star which is less than or equals to F X star Y, this is true for every X and Y. Now this proves X star Y star Y X star is a saddle point equilibrium. So this is a one very simple proof here in this proof we have used the fact that these functions are all convex they are continuous functions on a compact set and hence uniform continuity and we have used this thing. Now so far we have used we have assumed that F is strict concave and strict concave in the appropriate variables. Now what if they are not strict? If they are not strict then we make a perturbation here. So for example we consider epsilon XY to be F XY plus epsilon mod X square plus epsilon mod Y square we need to take here minus to make it concave and here it is plus. Now this becomes strict concave, strict convex function. Now there is a saddle point equilibrium for the game F epsilon S1, S2. So let this be X epsilon Y epsilon therefore F epsilon X epsilon Y epsilon let me erase this XY epsilon this is less than or equals to F epsilon X epsilon Y epsilon which is less than or equals to F epsilon X epsilon Y for all X comma Y in S1 cross S2. Now the that this sequence X epsilon Y epsilon has a convergent subsequence in S1 cross S2. So this extracting this convergent subsequence follows from the Boltzmann speed stress property S1 cross S2 are compact and hence this happens. Now let X epsilon Y epsilon converges to X star Y star along some subsequence. Now along this subsequence epsilon going to 0 in the equation star that we call this as star. So if I let epsilon goes to 0 we need to show that this converges to F X Y star this converges to F X star Y star this converges to F X star Y. This follows directly from this one F epsilon XY as epsilon goes to 0 this term and this term go to 0 and hence this follows so we get F XY star which is less than or equals to F X star Y star which is less than or equals to F X from XY in S1 cross S2 this proves the theorem. So what we have proved here is let us recall the statement if S1 and S2 are compact and convex subsets of some Euclidean space and F is a concave convex function then the saddle point equilibrium exists. Let us recall what we have proved under the assumption that S1 and S2 are compact and convex subsets and F is a concave convex function we showed that a saddle point equilibrium exists. Now there are few points in this proof did we really use the Euclidean space structure here. The answer to this is no because we only used the fact that they are compact and convex. So the extraction of all these things we only have used that under because these sequences are all from a compact set so they will have a convergent subsequence this follows because of by the compactness. So therefore this proof even though we stated it for Euclidean space it goes beyond the Euclidean space. So in that sense this is this theorem is valid even in an infinite dimensions as long as S1 and S2 are compact and convex sets and even the uniform continuity does not depend on R and SH it depends only on the compactness any continuous function on a compact subset is a uniformly continuous. So therefore this theorem is valid even in infinite dimension. For a matrix game if A is our matrix game or pi is the payoff function this is the mixed extension then this pi is now a payoff function from the mixed strategies of player 1 cross mixed strategies of player 2 to R and note that this pi is a bilinear function also delta 1, delta 2 these are all compact convex sets therefore mixed equilibrium exists. So this is basically the phoneme and min max theorem. So of course phoneme and spruf does not depend on that infinite dimension structure he has explicitly solved it for the matrix game case. So what we will do in the next session is to give a proof which works with matrix games and then we derive several other interesting consequences. We stop this session here and we will come back in the next session. Good day.