 So, last time we looked at two different ways of randomization in a dynamic game, particularly in a multi-act game and the two different ways of randomization were behavioral strategies, mixed strategies. Mixed strategy was a random choice of a pure strategy, randomly chosen pure strategy. So, you randomize over the set of pure strategies, so you randomly choose pure strategy and the behavioral strategy was that you at each information set choose an action at random. Now, and what we saw what was that these two can actually be from very different spaces, so there is you cannot directly talk of some kind of interchangeability between them. The number of parameters that define a behavioral strategy and a mixed strategy could be very different, but we asked for what is called equivalence. Equivalence was that regardless of the pure or the behavioral or mixed strategy played by other players, a behavioral strategy for player i was equivalent to a mixed strategy of player i if you had the following. So, the probability of reaching a node x under behavioral strategy bi while others play behavioral or mixed strategy mu minus i was equal to the probability that of reaching that node x when player i plays mixed strategy bi and others play mu minus i. This has to be true for all mu minus i and for all nodes x. So, in short regardless of the behavioral strategy or mixed strategy that others play the probability that you reach a node x has to be the same under the behavioral strategy of player i and under the mixed strategy of player i. Then the behavioral and mixed strategies bi and sigma i are said to be equivalent. And what we saw was that there are examples of either kind, an example where for every behavioral strategy there need not exist a mixed strategy and for every mixed strategy there need not exist a behavioral strategy. So, this is what we ended with last time. So, I also defined for you what is called a game of perfect recall and essentially a game of perfect recall was this had the property that. So, player i had perfect recall if you had this property firstly that two nodes cannot follow each other in the same information set. So, if they are in if two nodes are in the same information set then there cannot be a path leading from one node to the other. Secondly, if there are two nodes in the same information set let us say x, x dash and x double dash these are in the same information set. And then there is a if x is the is a predecessor of x dash then there must also exist a predecessor x hat of x double dash that is in the same information set as x and the action that you take at x and leading to x dash and the action that you take at x hat leading to x double dash should be the same. You can go back to the notes of the previous class to see the formal definition. So, basically what this meant was that the player did not forget that he acted he does not forget what he knew and he does not forget the action that is taken all of this. So, if a player is able to recall all of this. So, what I will discuss today is what is called Kuhn's theorem. So, these are two there are two parts to this. This Kuhn is the same as the Kuhn from Kuhn Tucker conditions or Karush Kuhn Tucker conditions in optimization. So, Kuhn's theorem basically gives us conditions under which behavioral strategies and mixed strategies are equivalent and you will see that the conditions have something eventually to do with memory. So, the first part of Kuhn's theorem or first of Kuhn's theorem let us say is concerns when does every behavioral strategy have an equivalent mixed strategy. So, this is for every behavioral strategy as an equivalent mixed strategy. So, this is what this is the case that this theorem this this theorem is concerned with. So, the theorem is this suppose each path from root to leaf to a leaf node intersects an information set of player i at most once then every behavioral strategy of player i has an equivalent mixed strategy. So, this is the this is the statement. Now, if you remember the example that we took where the there was a behavioral strategy that did not have an equivalent mixed strategy what was that example do you remember that what the example was. So, the example was something like this. So, the example was of this absent minded driver he had he took. So, he start you start from this node here and the both these nodes are in the same information set. So, when player when the driver takes a left turn he does not recall that he has taken left turn he gets the game moves on to the next node and then again he has the same set of choices. However, if he takes the right turn he the game ends. So, in this case what we saw was that it is impossible for us to get the outcome. So, I think we had named these outcomes O1 O2 O3 and we saw what we saw was that it is impossible to reach O2 for the driver to ever reach O2 with any with any pure strategy because a pure strategy would mean that he plays either L he has only two pure strategies either L or R if he plays L here then he come at this at this information set then he plays what that means is he is playing L and then again playing L because he is in the same information set and that brings him to O1. If he is playing R then he goes to O3 directly right. So, there is no way he can reach O2. So, in short that it is it is impossible to reach O2 using pure strategies and therefore using mixed strategies also. However, it was possible to reach O2 using a different form of randomization which is through big behavioral strategies right. This was the example and it turns out that this is only this is the only case where this fails. So, what that what Kunz theorem says is that suppose if you if you have if you make sure this does not happen which is that there is there is you cannot have a case where there is a path from a root load root node to the leaf node that passes through an information set twice. So, that intersects an information set of player i more than once then there is a then then every behavioral strategy of player i will have an equivalent mixed strategy. So, let us let us prove this now. So, first I think before we get on to proving any of this first time you should have need to have clarity about how do we compute the probability of reaching a node X under any strategy combination that players could play. So, mu is a combination of mixed or behavioral strategies of all players and I am talking I want to ask what is the probability that you reach a node X. Let us go let us go step by step and set up some notation. Suppose mu is any combination of combination of behavioral of mix. Combination means some players may be playing behavioral some players may be playing mix it is just a notation for that. Players can play whatever they want. So, we are considering some combination like this. So, first to begin with let us first set up some notation. So, suppose X is some node X is suppose some node this is some node X and then on the path from X to some leaf node there is another node X hat. Now X is suppose a node of player I now player I if you look at the path there is a path going from X to X hat and then further down to you know to some leaf node. Then player I takes and has several actions at X but there will be a unique action that leads you to X hat has to be because it is after all a path it has to be there is a one specific action which takes you to X hat. So, that action let us cause let us use that let us denote that action AI of X arrow X hat. So, this is the action at X leading to X hat. Suppose I take some node X here which is where we want to compute this probability. Now suppose player I so now this is let us say the root node. Now player I will play at several nodes on the way from root to this node X along this path. Now let that path be denoted by let suppose there are say Li X many nodes. So, these many equal to nodes equal to let us say equal to number of nodes from root from root to X and the path actual path actual nodes along the path from root to X where player I plays let us denote them by this. So, X I 1 to X I Li X. So, this is nodes on the path from root to X where player I acts on which player I plays on which P I acts. So, these are the nodes on the path from root root to X where player I acts. Now one thing to note here is see these nodes. So, X itself need not be a node where player I acts X is just some node in the tree where we want to compute this probability. So, question these are just the nodes along this path where player I is acting. So, this is the history of the game along which player I has is the state history of the game along which player I has played. Someone else had a question sorry about no it is not an assumption it is see it is it follows from because it is a tree you know. So, from X to X hat when you are going from X to X hat right. So, suppose here is X and here is X hat then you there is there is exactly one action that you can take to go from X to X hat no not immediate next not necessary see but it is a tree. So, it has to be that there is a unique action which takes you to X hat because it is a tree see if there are if there is more than one action suppose you could go like this way and then this way and then this way and then this way then it is not a tree anymore. So, it has to be there has to be a unique action. So, let us come to this let us first write out this this term this is the probability of reaching node X when players play some strategies randomized strategies mu. Now, can you decompose this in some way can it be decomposed let us say along the history or the path that leads from root to X this is all this is this comprises of all this is mu 1 to mu n. So, some may be playing mix some may be playing behavioral here from the root when everyone is played playing mu 1 to mu 1. Yeah, but then there is a sequence of action. So, the probability of first let us think in fundamentals. So, what is the probability of reaching a node X well to reach that node you have to take a sequence of actions you means all the players have to take a sequence of actions. So, it is equal to the probability that the first action is taken second action is taken third action is taken etcetera. So, the probability that the entire path from the root to X is is traversed with those actions that is that is that much is clear. Now, so this is therefore the probability that or in short all actions to X from the root are taken under mu. Now, so this is the probability that all such all the actions leading from root to X are taken. Now, so therefore, this is a probability of this is the probability of the end of several events you and first action is taken second action is taken etcetera. Now, does this decompose in any way? Does it decompose over the actions? Can we say the following? So, here is the basic question is this equal to the product of the probability of taking action 1. So, let us say probability of taking action 1 times probability of taking action 2 etcetera why are they not independent? Why what does that do? Very good. So, this is important see I cannot decompose this over the actions and the reason for that is these mu can be both mixed and behavioral strategies. So, what if it is a mixed strategy for example, so what mixed strategy will tell you for every information set what the what action is to be taken right that is a one specification sorry a pure strategy will tell you for every information set what action is to be taken. A mixed strategy is just a random choice of a pure strategy. So, suppose this pure strategy is chosen the one that I have marked this right pure strategy which chooses this action here and this action here. Then the probability of choose and suppose these two actions are on the path leading to X then the probability of choosing these two actions let us say action a 1 and action a 2 is equal to the probability of choosing that mixed strategy. You cannot multiply this times this because probability of choosing a 1 and a 2 is the probability of choosing that mixed strategy. So, when you choose a mixed strategy all these actions get chosen together is that clear? This product is basically will end up double count or double multiplying essentially the same probability you will write something for a 1 then you will write the same term again for a 2 whereas a 1 and a 2 are being chosen together if there is a mixed strategy is this clear? So, in short we cannot decompose this along the actions in that path. But we can decompose over players because after all actions are being chosen some set of actions are being chosen the actions chosen by player 1 are chosen independently of the actions chosen by player 2 and so on is this clear? So, you cannot write this as the following. So, the probability that player i chooses actions leading to X under his own strategy now mu i and the product over this from 1 to n. So, the actions of player the actions that player i have has chosen they are the probability with which those are being chosen or is independent of the probability with which the other players are choosing the. So, that event is independent of the choices of the other players. So, there is this decomposition therefore over players. All right. So, good. So, now essentially what we are concerned about therefore is this we need this we need to get a handle on this one the probability of choosing actions leading to a node by a particular player considering that considering whether he is playing a big strategy or a behavioral strategy. So, that is what we are going to come to. So, now let me ask you this. So, I will let us set up a notation for this. Let us this particular thing this component of it this contribution let us call this probability of p i of X given sorry mu i. So, this is that this is the component this sorry I should probably use some other not let me use problem instead of problem p. This is the contribution of player i's thing in this whole sum. So, prob i of X given mu i is the probability of taking actions leading to X by player i when he is randomizing using strategy mu i. Is this clear? Okay. All right. So, now let us take be more specific suppose suppose suppose player i plays let us say a mixed strategy sigma i a mixed strategy sigma i then what is this probability marginal what marginal no I want to express this now in terms of sigma i I am telling you he is playing a mixed strategy. Okay. So, let us write out some notation. So, remember gamma i is the set of pure strategies of player i and let us write gamma i of X okay. Is this is the set of pure strategies of player i gamma i of X is the set of pure strategies of player i where player i takes actions leading to X okay. Now, where does you take these actions leading to X remember this is these actions are being chosen at these nodes right this is these were the nodes on the path from root to X where player i was acting at these nodes he has to take those particular actions okay. So, the pure strategies that prescribe those actions at those nodes are all included in this set gamma i of X clear okay. All right.