 So, now can you tell me now I have some notation with so gamma i of x is there so can you tell me what is then this prob i of probability sub i of x given sigma i in this gamma i of x. So, it is equal to sigma i of gamma i where gamma i belongs to gamma i of x. So, let us absorb this properly. So, this here is the probability of choosing a pure strategy gamma i. This summation is over all pure strategies gamma i where the action leading to x is being chosen and that is exactly this term, okay, all right, good, okay, okay next. So, what if it was a behavioral strategy now? So, suppose, suppose player i plays behavioral strategy bi then can you tell me what is problem this? So, if he is playing a behavioral strategy bi then it means that along all along that entire this this whole place here along in all these nodes he is again going to be taking the exact same actions that we just mentioned, okay. But then he how is he randomizing? Every time he gets to an information set he chooses an action at random, okay and sort of implicit in all this is that this randomization is being done afresh means independently at each information set, okay. So, there is independently tosses a coin and chooses an action at each information set which means that essentially this therefore what is this equal to? This is the probability of choosing the first action times the probability of choosing the second action times the probability of choosing the third action etc because at each information set he is randomizing in separately, right. So, there is an independence across the time epochs, okay. So, therefore, this will be the product of first let us write this. So, this is the behavioral strategy of player i the probability with which he chooses an the action leading to x, okay. So, suppose he is at a node xi so the suppose he is at a node xi l small l the action that he chooses on the path leading to x is a of this arrow x, right this is this was our notation this was our notation here, okay. So, a is the action at x leading to xi, okay. So, we want the action. So, what this means is when he is reached the node xi l is going to take the action a i of xi l arrow x a i of xi l arrow x at node xi l, clear. This here this this specifies the gamma i is yes the in the sense that yeah no no because on all other information sets the player can do anything. So, on these information sets these actions are being taken correct that is right, yeah, okay. And remember this behavioral strategy is this action given information set right and let us write this I will have another notation here, okay. So, I now what is the product over yeah l going from 1 to Li x, okay and this here is the info set containing xi l, okay. Now, technically I should be careful here this is if Li x is positive, okay. The if I like this is if Li x is positive that means if there is a path actually where from x if he if player i does play along this path, okay. If he does play along this path, if he does not have any information set on this path if then he is not even playing on this path that means Li x then is 0, okay. If Li x is 0 then what should be this probability 1 it is because it is his contribution to that product, right. He does not he basically has nothing and no action. So, the probability for that the probabilities is 1, okay, all right. So, now you have clarity, okay. So, how do we compute the probability of reaching node x of player i taking actions leading up to x under behavioral as well as under mixed strategies, is this clear, okay. So, now what we want to do is we want to let us come back to the statement of our theorem. The statement was for every behavioral strategy has an there is an equivalent mixed strategy. So, what we will now do is we will take an arbitrary behavioral strategy construct from there a mixed strategy and show that the two are equivalent, okay. Now, to show that to the two are equivalent we will have to show that the probability of reaching an arbitrary node like this is the same under behavioral or mixed, under the two strategies regardless of what the others are playing. Now, the regardless of what others are playing becomes very easy here because remember this whole thing, this expression here you see what happened to this expression. This expression actually became a product over players. So, the components contributed by the other players are going to be the same in that equation. So, when I write out this each of these is actually a product over the players, right. So, this is sorry, this times and that this is on the left hand side and same thing on the right hand side, right on the right hand side I will have this, this here and this here is the same, right. Because this is when with player i playing bi and others playing mu minus i that thing becomes this left hand side with player i playing sigma i and others playing mu minus i you get the term on the right hand side. But the and the reason you get this product is because the players are randomizing independently. So, you got this decomposed over the players, right. So, you have player i's contribution here and this is the all the other players contribution. This is player i's contribution, this is all the other players contribution. But all the other players contribution is going to cancel out from both sides. So, all I care about is player i's contribution, okay. This is remarkably elegant actually because when we set this up, we said that this is a looks like a very demanding criterion. Regardless of what others play i should be able to reach the node with the same probability, but actually in fact it does not you know that is how it should be. Because you know you do not want to when you so when you set it up this way, it actually it actually becomes very clean because what the other players play is can be held can be cancelled out from both side. So, this part has to therefore become has to be somehow shown to be equal to this one through a suitable construction, okay. So, what we are going to do is we will start with a behavioral strategy bi construct a corresponding sigma i and show that this term maybe I will write this term is equal to this term that is going to be the agenda, alright. So, now here is the so let so now for the proof let bi be a behavioral strategy of player i, okay. Now I want to construct a mixed strategy, okay. So, sigma i of gamma i this is the now I am going to construct a mixed strategy, sigma i and so what means that for every pure strategy gamma i I need to give you a probability with which that pure strategy is being chosen. So, the probability is like this, see you constructed like this you do this suppose you first let us write this. So, this strategy pure strategy gamma i specifies an action for player i at each information set. So, at information set eta i it is it specifies the action gamma i of eta i this is the action that this is to be chosen at this information set, okay. Now you look at the probability that you choose this action under the behavioral strategy bi okay which means what is that probability that is equal to this bi of gamma i of eta i given eta i and what you do is here is the very interesting thing you do a product over all information sets of player i, okay. So, a pure strategy is a sequence of actions and what you what he is basically doing what he is doing is he is saying let us take the probability with which each of those actions is being chosen under the behavioral strategy take the product of all of those, okay. That is the probability of choosing that pure strategy under. So, it specifies a mixed strategy and gives you the probability of choosing that particular pure strategy, okay. Now question is firstly why is this even a probability this is the first thing we need to establish right you can give some formula but why is this even a probability. So, what we need to first check is sigma i greater than equal to 0 and does it satisfy this which is so sigma i greater than equal to 0 is trivial of course it is but this is what needs to be shown I need to show that this is in fact of a probability and this sums to 1, okay. So, what we need to show is first this you take this product this is the probability of choosing pure strategy gamma i sum this over all gamma i in the set of all pure strategies and what we want to show is that this is equal to 1. So, this looks horrendous in the in its in the way it is written and all that but there is actually a lot again very it is super elegant due to various reasons, okay. So, I will so let us let us understand this. So, this is the set of all pure strategies can you express this in terms of information sets and actions I keep asking you this right how many pure strategies does the player have how do you find out how many pure strategies. So, if you have say suppose k information sets and you have m actions and m actions at each information set how many pure strategies does the player have m power k right. So, how did you get m power k you take the number of actions at each information set multiply multiply multiply right. So, that and you multiply k times you get that. So, actually this same this basically gives us a way of enumerating or of describing the set of all pure strategies. See the set of all pure strategies is essentially equivalent to taking the product of the action sets at over all information sets. It is essentially a long tuple of actions the length of the tuple being equal to the number of information sets. So, I can do the following I this let me write it out then it will be clear this the set of all pure strategies is in fact the same as this what was my notation for actions at an information set I think. So, let me write like this u i. So, this is the set of actions of player i at information set eta i. So, what am I doing here I am taking a Cartesian product of the action sets every element of this whole Cartesian product tells me this action from this information set this action from this information set etc. It gives me a basically one action per information set and that is a pure strategy right. So, the set of all pure strategies is actually equivalent to this product essentially the product of all the action sets. So, therefore what this means is this summation that we have here can be written like this I can write something like this I can write a i 1 in u i eta i 1 a a i 2 in u i eta i 2 dot dot dot. Let us say there are k information sets then a i k u i eta i k k is the this times then my this thing times the product over now I can do a product over let us say L equal to 1 to k B i of now what is what is B i what do I need to write in place of B i well B in place of B i essentially what I am now talking of taking action a i a i L at information set eta i L right. So, I am writing a i L eta i L. So, this is the first action this is an action from information set from the first information set several actions this because a i is i is for player i one is the first information set. So, a i 1 ranges over this set yes yeah because I am summing see I have to take the product of all these sets. So, I range over every so it is like a product of it is your summing over a large tuple of variables first you sum up the first component then you sum up the second etcetera it is essentially a very very very complicated multi sum here times this product times this whole product. But once you see this this actually becomes very easy. So, essentially now all I need to do is well this here is there is a product here there is a product there is some outside. So, what I can do is I can take the k th term inside and sum over only a i k ok. So, let us look at the k th guy here that k th guy here depends on a i k. So, the summation over a i k can go inside and the rest goes out comes outside. But so what is going to happen is I have a summation over a i k in u i eta i k I have b i of a i k eta i k alright and then outside I have my product now going from l equal to 1 to k minus 1 b i of the same thing and then and here I have my summation over a i 1 all the way till a i a i 1 in u i eta i 1 all the way till a i k minus 1 in u i eta i k minus 1. So, now what is this? This is the sum over all actions the behavioural the probability of taking that action summed over all actions this whole thing is actually 1. So, this no there is more to it because of the structure of the problem. I mean it is the so this identification is important that you can identify the set of pure strategies with the set of actions of course, this is just a this is okay this is just a joint distribution which I am going to say well it has to yeah I mean I am just doing the product of marginals or whatever alright and then therefore this guy sums to one and so this whole thing will telescope and you will get finally the whole thing will become one okay. So, what has this shown? This is basically shown this claim here that sigma constructed this way is now a valid mixed strategy okay. We have not yet shown that it is equivalent to b i okay to show that it is equivalent to b i we now need to argue that this is equal to for all points for all nodes x okay alright. So, let us begin with the left hand side. So, left hand side is is this okay by the way I forgot to mention I was a little bit of a hurry okay. So, where did we use that that the path from root to x passes through an information set only once which product which this one this is just a construction this is this is just my construction I mean I am free to define it whatever the one above okay. So, the fact that I have written this here is a subtlety basically I wrote out a path I actually wrote out or rather I wrote the nodes along which the player plays that is x 1 x 1 whatever what was this yeah x i 1 to x i li x right this is the these are the nodes at which the player plays the nodes at which the player plays will be equal to the number of information sets provided each node provided the path intersects each information set at most once or exactly once right. So, so if they if this is the nodes at which the player plays now if an information set passes through two nodes in which on which the player is playing then this this product cannot be tagged to the number of nodes it will have the this product is really over the number of information sets but we have enumerated it over number of nodes and that is because every information set is intersecting the path exactly once when a player is playing it is playing at each of these nodes he is playing in a distinct information set each time is this clear. So, this is where that has been assumed sorry I am I in my in hurry to do this I have actually I forgot to point this out okay. So, is this clear so this is where we have made the assumption that the path every path intersects every information set exactly once okay or at most once okay all right okay. So, now let us come back to this so now we want we have this probability this probability we said is equal to is equal to sum over gamma i in capital gamma i of x sigma i of gamma i now sigma i of gamma i I we just have a construction for that which is the product over eta i which is overall information sets of B i of gamma i of eta i given eta i okay. So, now let us so i i is the set of all information sets of player i this let us write it in the following way let us write it as i i 1 union i i 2 okay this here these are the information sets on intersecting the path from root to x okay. So, these are the information sets intersecting the path from root to x okay and these are the other information sets okay then in that case I have summation gamma i in gamma i of x and then this product the product then can be written as you have one term which is i 1 and now what is this this was actually the same as what is this this is the probability of taking the these actions along probability of taking actions leading to x okay under the behavioral strategy B i. So, this is nothing but my problem okay. So, and this term here the second term then just remains so this guy and okay the first so this is probability of x given B i so this has nothing to do with gamma i so this will come out and what I am left with is then this is this probability this summation is over all gamma is which take actions leading leading to node x okay. So, what is this summation of equal to what is this whole thing equal to no it is equal to now the probability under B i right probability of reaching node x under under B i see this is exactly the same actions that you would need to take under but under behave under the behavioral strategy B i okay. So, the so if this that is why this now becomes independent of the specific choice of the pure strategy because you though the those in fact that answers this question also see the this automatically specifies for me the the sequence of pure actions that I need to take right. So, it has it has told me what so so that part of my pure strategy is already specified and it is fixed in fact I have to take those actions in all my every function in gamma i of x every strategy in gamma i of x takes exactly those actions okay. The probability of taking those actions under the behavioral strategy is here all right now what what is this summation then here I am allowed to take other actions that other information sets which do not which are not along the path from root to x there I can do whatever I want right. So, in short this is this is nothing is in fact like exactly like this summation it is exactly like this summation but on the remaining part of the tree where which is not going from root to x. So, this thing actually sums to 1 okay because the set of all that is actually the so the this is over gamma i in gamma i of x it is actually not gamma i of x that restriction does not make a difference to this this summation because anyway this is the the information sets involved here are not currently constrained in the choice gamma i of x okay. So, this whole thing therefore becomes equal to just from this all right. So, this so this basically has shown that your the for if you have if an information if our path from root to node root to this thing intersects each information set exactly once or at most once then every behavioral strategy will have a equivalent mixed strategy is equivalent to sigma i.