 We discussed a result, a theorem in the last module which gives us the necessary and sufficient condition when we can have a mixed strategy equivalent of a behavioral strategy. And that theorem was this one, so it says that the behavioral strategy has an equivalent mixed strategy if and only if each information set for that player intersects with every path starting from the root at most ones. So you cannot have situations like where the same information set is intersecting with the same path multiple times as we have seen in that example which represented some amount of forgetfulness. Now in this module we are going to discuss the opposite that when can we have a behavioral strategy equivalent of a mixed strategy and we will finally see that we will give a sufficient condition for that and that sufficient condition will also subsume the condition that we have given in this previous theorem. So therefore we can have an equivalence between mixed strategy and behavioral strategy. So with those conditions that we are going to state you can have a behavioral strategy equivalent of mixed strategy as well as mixed strategy equivalence of a behavioral strategy. So these two things will be one and the same. So in order to define that we will have to somehow capture what is forgetfulness of a player. So we have seen a couple of examples for forgetfulness of the player one in which the player forgets whether it has played at all or not and the other condition was where the other situation was where the player has forgotten which action it has played. So we will address both of this forgetfulness one by one but before that we need to define something we need to define what is a choice of same action at an information set. As you can imagine the name itself is quite explicit in saying what it means. So it is saying that if you have two different vertices so let's say I have two different paths inside a network so let's say one path is coming like this from root to here and another path is possibly coming from the same root to some other node. Now what it is saying is that if you intersect any information set so let's say you have two different nodes let's say xk and xl right. So these are the two points at which this path these two different paths let's say this path is x and this path is x hat they are intersecting with this information set of player i let's say this is player i's jth information set then we say that these two different paths choose the same action at this information set iij if of course k is less than the length of the path and also l is less than the length of that path and the action that you are picking so for instance if you are at this node which is inside this information set and you are taking this action to go towards the end of this path so going towards the next node in this path then they should be the same so if you are taking the same action because in the same information set you have the same set of actions available to you so if this path so let's say here you are taking a one then it must be the case that here also you are taking a one to go to the next level the next node in that in that path then we are going to say that these two paths are choosing the same action at an information set of iij this is just formally stating what it means but the but the name itself choice of the same action at an information set is already saying what it is what it is doing now I just want to make a remark that even though in this particular case we are just looking at the path which is from xk to xk plus one which means that it is the very next node in the same path but it does not always so this this direction and the corresponding statement that xk leading to xk plus one does not always necessarily mean that it is just the next node in the same path so I can have let's say I have a bunch of nodes in a specific path and I am just talking about let's say this one is x and this one is y and there are some intermediate nodes in between so I can say that this action is leading to so let's say this is the action a prime which is leading to y so I can say that the action for player i which which starts so which takes this path from x towards y this will be equal to a prime so that is that is the meaning of this statement that I can have a path which is leading to y from x but that that y need not be the immediate next node so we will be using this term leading to sometime later in this in this module so therefore it's worth noting what it means alright so now we have let let us go to that particular condition where we can have this equivalence or at least we can say that you can have a behavioral strategy equivalent of a mixed strategy and that condition is what is known as perfect recall again the name itself somewhat resembling what it means essentially essentially the definition rules out all those two cases that we have discussed in the example before where people that the player was forgetting which action it has played and it was forgetting whether it has played at all or not so let's look at the first condition so we'll say that player i has perfect recall if this two conditions hold the first one is that every information set of player i intersects every path from the root to a leaf at most once so this is this is actually ruling out the second kind of forgetfulness very you had so remember we had an example where these two nodes which were actually connecting a path from from root to root to the leaf and they were connected with a information set so this condition one is actually ruling out that kind of situation because this information set is intersecting with the same path multiple times and that is not possible according to one and the second condition says that every two paths that end in the same information set of player i so again you can draw the picture that you have two different paths that end in the same information set for player i so let's say your eye has a specific information set iij and you are you are looking at these two nodes which are coming via different paths and they should so every two such paths that end in the same information set of player i pass through the same information sets of i so it might should not be the case that they pass through different information sets in the same order so you cannot have arbitrary orders so you you will be sequentially going through the all the information sets so if there are other information sets let's say j prime and more information sets of that player then all these things all these two paths will actually be intersecting through all of them in the same sequence and then and in every such information set the two paths choose the same action so this is the most important thing so if we because in the same information set the set of actions are available to this player player i is the same they should also choose the same action so this is actually ruling out that particular example where the player the same player were playing both left and right and then it is it was in a situation where it had a information set there so it means that it is taking so here there was an information set so here also player one was playing here also player one was playing so here player one was taking two different actions and these two paths were actually leading to the information set where this agent of the same information set of that player so this is ruled out by this second condition because now we are ensuring that it not only takes the same sequence of information sets of the same player it also takes the same action so this cannot happen that it is taking different actions and leading to the same information set okay so this is this is essentially the definition very carefully ruling out this this kind of forgetfulness situations so one can write this a little more formally and this will this will require a little bit of time to understand this this notation so let let us go over it very slowly so for every i i j so this is an information set of player i and every pair of vertices in that in that information set let's say x and x prime so something similar to this one we'll just go back to the into the same diagram back and forth so if the decision vertices of i so note that these are decision vertices of i so those are the places where player i is essentially the player so this sequence of of nodes are only those places where player i is playing so therefore these are only those places where you have play the information sets of player i and there could be some intermediate nodes so these are not sequential nodes in this in the path because there are some more nodes where some other players are playing but we are just picking out only those points where player i is playing so let's create the sequence of those those nodes where player i is playing with x i 1 x i 2 and finally the the lth node is the node of x that we are comparing with so this is x and this is x prime so you have two different two different paths where these players are actually playing so the same player i is playing and this has a if you list them down there are l such points capital l such points in this path which is leading to x there are k such so l prime such nodes which is leading to this other other node x prime then three conditions should should get satisfied the first condition is this this number l and l prime should be the same so this is the first condition here so you cannot have this similar number of dissimilar number of points where player i is playing and you are leading to the same information set for that player that is not possible the second condition that should hold is that for if you look at any intermediate point in in this sequence of nodes so let's say look at l and the corresponding l here then they should belong to the same information set for player i so we are going sequentially one by one and listing only those points where player i is playing and we have ended up in the in an information set in the same information set for player i then if you look back any previous node so if you are looking at the same numbered node in both these two sequences they better belong to the same information set for that player and the last condition is the action leading from l to l plus one so here you can see that l and l plus one they need not be a parent and child relation so they might there might be in some intermediate nodes in between but because this this is a game tree so this path which which is leading from xi to xil to xil plus one will be unique and the action that is taken by this player at that point for both these nodes so xil to xil plus one and xi prime l to xi prime l plus one these two actions should be the same so this is essentially capturing those two conditions that are that I have said in words in the here so that is being formally written with the notation that we have used so far so we will also call a game to be of a perfect recall the entire game to be having perfect recall and naturally if every player has a perfect recall so this definition as we have already mentioned the the first condition essentially is subsuming the condition of of that theorem of the previous theorem that we shown before so so therefore I mean there that condition was necessary and sufficient for every behavioral strategy to have a mixed strategy equivalent mixed strategy now we will show that with this perfect recall definition which is a more strict definition than that because we already have that that condition as well as we have the second condition which is also added on top of that so it is a more restrictive definition than the previous one we will see that we have equivalence in both directions so let us look at some examples to understand what this game with perfect recall mean so this example if you look at so there are three players player three is playing first and based on its two actions either one can play or two can play then based on their actions you can come back to one again and based on player one's action there could be such this kind of situation so why is this example of a this example is that of a game with a perfect recall because if you look at those two statements that you have so let us look at the text statement so every information set for that player intersects with every path from the root it at most one so there is no such situation where the same information set is interacting intersecting with the same path from the root multiple places and the second condition was that there are two parts that ending ending the same information set they will pass through the same information sets of that same player in the same order and in every such information set these two parts choose the same action so of course this second part we cannot verify because there is only one one situation so there is no previous information set of either one or two but at least we can see that the first condition is getting satisfied and also the second condition is sort of backwardsly satisfied but if you look at the so modify the game as slightly by by introducing this player so instead of player three now player one is actually making the first move then this is certainly violating the condition that it is it is not taking the same action so you see that it is taking two different actions but if you look at this nodes let us say this is x and x prime and if you look at the path starting from the root to this node in the previous information set of the same player which is this the player was taking two different actions one is taking this action the another was taking a different action to reach to these two different nodes in some sense as we have showed in the previous example as well this player player one is actually forgetting which action it has played so that is why it is in this information set it is forgetting what action it has picked in its in its previous round and that is not allowed according to the definition of perfect recall so let us also look at what are the some implications of perfect recall so the the perfect recall definition will also give rise to certain certain equal equality in this strategy sets so let's define this one so let's say si prime si star of x be the set of pure strategies of player i at which he chooses the actions leading to x so there might be multiple multiple actions so let's say we are at a specific node x at some some some node in this game tree and there are maybe multiple action multiple information sets which are actually leading to this x so maybe this this path and then there are there are other information sets of of the same player so let's say upi one b ii two and so on so this is ii j for instance and if you look at all these if you collect together all the actions that has actually taken has been taken by this player i to reach to this node x that those collection of all those actions or the strategies we will put as si star of x so these are the set of pure strategies of player i at which it chooses the action leading to x that is the intersections of this of the members of si with the path from root to x so then then we have this theorem it says that if i is a player with perfect recall and x and x prime are two vertices in the same information set of i so now x and x prime let us assume that it is also a member of some information set for player i then it must be the case that the the corresponding sets the corresponding sets that we that we have just defined for these two nodes x and x prime should also be the same so this conclusion essentially is a result of of that condition that it should come from the same sequence of information sets and you will you will take the same actions at those information sets so collecting all these points together we can actually now state the the theorem which is essentially a sufficiency theorem due to coon in 1957 so in every imperfect information extensive form game if i is a player with a perfect recall then for every mixed strategy of player i there exists a behavioral strategy so of course the converse is already true because in the game of perfect recall already considers the the necessary and sufficient condition where behavioral strategy has a equivalent mixed strategy that direction is if and only though here it is just a sufficient condition so together we can say it is safe to have a game with a perfect recall where we can use behavioral strategy and the mixed strategy interchangeably so i'm not going to prove this this is straight from the book of masler and this theorem 6.15 can be can be read in order to find the find the corresponding proof the proof essentially let me give you the intuition it's a constructive proof it starts with a mixed strategy and essentially constructs the behavioral strategies such that the probabilities of reaching a leaf uh reaching every leaf node are the same and this arguments essentially uses the fact that we have defined the definitions of perfect recall that i mean such kind of an construction is always possible because it's a game of a perfect recall