 So, now we have to ask the reverse question which is for every mixed strategy, does there exist an equivalent behavioral strategy? So, let us first do a simple lemma, so if player i has perfect recall, then for any two nodes x and x dash in the same information set, what can you say about this, so but can you tell me why, so player has perfect recall means what, so these are what is gamma i of x it is the set of pure strategies in which a player i takes actions leading to node x, now x dash is another node, but it lies in the same information set as x, in the same information set of player i, so these two are in the same information set of player i, now if these two nodes are in the same information set then gamma i of x is equal to gamma i of x dash which means the set of strategies that get you to these two nodes has to be the same, why is that, go back to the definition of perfect recall, see perfect recall was asking that if two nodes are in the same information set and one of them has a precedent predecessor, so x and x dash are in the same information set, x has a predecessor, let us call it x bar then x dash will have a predecessor x hat suppose and x bar and x hat will be in the same information set and moreover the action that you take here going leading up to x has to be the same as the action that you take leading up to x dash, what does this mean, that means if x and x dash are in the same information set of player i, you look at any preceding information set in which say x has some predecessor here, the action that you would take to lead go up to x is the same as you would take the as you would at x hat, so in short this same thing now I can keep repeating all the way back, so not just for this predecessor all the other predecessors of x also, I will have the same problem essentially, so what this means is the fact that I am going to x and the fact that I am going to x dash locks my actions essentially I have the, if x and x dash are the same you are in the same information set from root down to here player i should have taken the same sequence of actions to get to these two, so consequently it means that as far as the path going from x, going from root to x is concerned and the path going from root to x dash is concerned the same sequence of actions are being chosen, on other paths the player can do whatever he wants, so therefore this essentially is ensuring that gamma i of x is equal to gamma i of x dash, the set of all such pure strategies is actually the same. So, this is quick lemma, so now let us write out the theorem and hopefully let us see if we can prove this today. If player i has perfect recall then for every mixed strategy of player i there exists an equivalent behavioral strategy, so for the proof, so first let us set up some more notation, so suppose for any node x and action let us say a in let x superscript a be the game reaches a is played at x, so suppose you have a node and you have an action that is available at that node, so node x and an action a that is available at that node then when the player takes action a the node that the game reaches is denoted by x a, x superscript a, so now what we want to do is we want to start off with a mixed strategy and show that there is an equivalent behavioral strategy, one other little thing related to this let me write this out, so can you tell me what is this, so if a is an action that player i can take at an information set eta i of x then what is gamma i of x superscript a, what is the set of pure strategies can you express this in terms of gamma i, x superscript a is this node that the game reaches when a is played at x, so can you write the set of pure strategies in which player i takes actions leading up to node leading to node x a, now express this in terms of gamma i of x action a at x right exactly, so this is gamma i in capital gamma i of x such that gamma i at the information set eta i of x is equal to a, so all those pure strategies which specify this particular action at this information set all those pure strategies from gamma i of x which specify this particular action, so this will give us some structure, so now let, so step one, let sigma i be a mixed strategy of player i, now what we will do is we will define this in the following way, we will define a behavioral strategy, the probability of taking an action a at an information set eta i, this I am putting this notation like this because I want you to see that this is actually a conditional probability, so this is the probability of, this is is the probability of taking actions leading up to node x a under the mixed strategy sigma i divided by the probability of taking action, of taking actions leading up to node, so this is eta i and let, so here is how I am defining my behavioral strategy, so for any information set eta i I look at a node x in it and I do this, I take the ratio of the probability of by that the mixed strategy specifies of taking an action leading up to node x a, where a is the action of that, so for any information set eta i and an action a at that information set, the behavioral strategy I am defining is going to do this, it is going to give probability, the probability it is going to assign to action a is this, so it will take the, it is the ratio of the probability of taking an action leading up to node x a under the mixed strategy divided by the probability of taking an action leading up to node x, so it is essentially the conditional probability, given that you have reached node x, what is the probability now that you are taking action a, because this is the probability of reaching node x and then taking action a, this is the probability of reaching node x, is this clear? Okay, one subtlety here, I need to worry about what happens if the denominator is zero, say for example, there is no pure strategy, so rather there is a mixed strategy, the mixed strategy I am choosing does not give any probability to pure strategies that reach x, okay, so if the denominator is zero then what do I do, so for that, so this is therefore defined only for if denominator is greater than zero, now if the denominator is zero, then what you do is simply choose an action uniformly at random, so if this, if you are denominator which is this summation, if the denominator is equal to zero, okay, now here is at least we can complete this one part today which is this, so now I have just constructed a behavioral strategy from this mixed strategy, right, it specifies this as the probability of taking an action in a certain information set, now you do you see something odd in this, in this construction, so that is something very odd actually if you look at it, see here on the left hand side I have an information set and on the right hand side I have actually a specific node in the information set which is x and I said I will let x be any node in this information set, so this is a well defined definition provided the right hand side is actually independent of x because eventually a behavioral strategy should act only on the information set not on the specific node, right, so it has to be invariant with the choice of the specific node in the information set, so I need to first show that this is in fact a well defined behavioral strategy because it has to be independent of the choice of x, so long as x is in eta i, is this clear, so why is this now independent of the specific choice of x, so that needs to be shown, so first to begin with for this to be a valid behavioral strategy, so first let us now take, let us take two x and x dash, so step two showing that bi is a valid behavioral strategy, so now if let suppose x, x dash are in the same information set, if x, x dash are in the same information set what did we just conclude, layer i has perfect recall, so we just concluded that we just concluded this, we also said that this particular way of expressing gamma i of xa, so consequently what happens is if you look at this denominator here, if I replace x by x dash this denominator is going to be in the same because gamma i of x is equal to gamma i of x dash and in fact numerator is also going to be the same, right, so this follows from the lemma, so therefore this in eta i then this follows from the lemma, all right, good, so now this means that it is in fact a function of just the information set, I also need to argue that this is in fact a behavioral strategy which means that if I sum this over a I should get 1, it is clearly greater than equal to 0, I need to show that also need to show that sum of bi of a given eta i a in ui eta i this is equal to 1, how do I argue that? That is actually again straightforward from the definition of gamma i of xa, we also need to show as I said we also need to show that this sums to 1 and the reason this sums to 1 for that all I need to argue is that the numerator when I sum, numerator depends on a, the denominator does not depend on a, here this numerator depends on a, denominator does not, if I sum the numerator I should get back the denominator, right, that is all I am looking for, so that is actually easy to argue basically what I have is remember if I have two distinct actions a and let us say a dash, so these are say two distinct actions then this is or rather why distinct actions if I in fact I do the following, this if I take the union of all of these over all a in my action set at the information set eta i this is a disjoint union equal to gamma i of x, so for distinct actions these are disjoint for distinct a, right, so if you take because after all at eta i I am asking you to take an action a, if the actions do not match then obviously the pure strategies will be this different, so this gamma i of x a actually if I take the union over a it is actually a disjoint union and that gives you gamma i of x, so that from here it follows that therefore that the if I sum the numerator over a this is in fact equal to the sum over this is in fact equal to the denominator itself, okay, all right, so this is just established that we have a valid behavioral strategy, okay, step 3 is this, so suppose x x i 1 to x i li x, suppose these are the nodes at which player i acts leading to x, now if I look at the let us say the l plus 1th node in this, this is actually nothing but the, so the pure strategies in which player i takes actions leading up to node x i l plus 1 is actually nothing but the pure strategies in which player i takes actions leading to node l and then takes action the action specific, the action that takes you from x i l to x i l plus 1, right, so this a where a is equal to a i of x i l going to x i l plus 1, okay, all right, now let us just calculate this probability and that is that will be it, so probability of reaching node x under a behavioral strategy bi, this probability is equal to the probability now of taking the required actions, the probability of taking action a i of this is the probability of taking action leading to node x at the information set eta i of x i l under the behavioral strategy bi where behavioral strategy bi was defined in this through this whole formula here, okay, through this two case by case formula, okay, so then this, so now, now there are actually two cases, one in which the denominator is 0, the other is denominator is not 0, let us just do the denominator not equal to 0 case because that is the more complicated one, the denominator equal to 0 is actually kind of degenerate and it works out easily, okay, so this therefore is summation of gamma i of capital gamma i, so let me denote this guy here by a l, okay, a l is the action, the action that you take at node l going to node l plus 1, so a l is here, okay, so this is a l, so this is just substitution so far, but notice that what is the set of pure strategies in which you take actions leading to node x i l plus 1, it is the pure strategies where you take actions leading to x i l and then take action a l at that node at x i l, so what is going to happen is you have actually this product now over l, but then the numerator and denominator of consecutive terms are going to be equal, you are going to get cancellations, so this here is the, for any l is actually going to be the numerator of the next term, right, because you keep on dividing by the probability of reaching a particular node and then in the next step, you have probability of reaching that node divided by the probability of reading the previous node etc, etc, so this actually telescopes and eventually what you are left with is equal to the probability, so you are going to be left with this summation, you are going to left with summation gamma i in capital gamma i of x because that is the last node you will reach, that is the last node divided by sum over gamma i of the probability of gamma i, let us say x i 1, now so what you are left with is this term where the numerator is the probability of taking actions leading up to the node x that you are concerned with, denominator is taking the probability of taking actions up to the first node where you have and what is that? That is actually the set of, this gamma i of x i 1 is actually the set of all pure strategies because this is the first time you have, right, so this is actually the set of all pure strategies, so the denominator is in fact, this whole thing is in fact equal to 1 and so therefore, you get that the numerator, so the probability of, so what we get is that the probability of reaching node x under behavioral strategy B i is equal to probability of reaching node x under sigma i, this remember is nothing but the probability of, so we get rather let us say we get this and this is nothing but this probability, that is all, so that is all there is to it, so this ensures that we have that for every mixed strategy, that is actually a behavioral strategy, so you can see the main thing we used is really this lemma to be honest, perfect we call what it ensures is basically this, you know that is all that is being used, all right.