 We spoke about a different kind of randomized strategy in the context of IIEFG in the previous module. So that randomized strategy is called the behavioral strategy and we have also seen what relationship it has with the mixed strategy that we are more comfortable with. We have discussed that in larger detail earlier. Now why is behavioral strategy desirable particularly in the case of extensive form games? It is more natural in larger IIEFGs. Players generally plan at a stage at an information set level rather than having a master plan over all the possible strategies starting at the very beginning if the game is long enough. And also from a computational viewpoint it is a smaller number of variables to deal with. So let's for an example pick a player who has four informational sets and in each information set it has two actions each. Then in order to describe the mixed strategies you will need some 2 to the power 4 minus 1 variables because you will have to first list down all possible strategy profiles for that player which will be 2 to the power 4 and because they will all sum to 1 the number of independent variables will be 2 to the power 4 minus 1. So this many number of variables and the numbers will increase as you have exponentially increase as you have more number of information sets. While for behavioral strategy you will only need 4 variables because at every round you are going to toss a coin and taking the action accordingly. So both from the point of view of the cognitive limits of the players and also from the point of view of computation behavioral strategy is more desirable. Now the question becomes moot if we have an equivalence for both these two type of strategies. If it was always possible to find a behavioral strategy for any game for any IIEFG for any kind of mixed strategy then we can equivalently talk about behavioral strategies on mixed strategies. But in this module we are going to discuss that it is not always true that you can construct one from the other for any kind of game. So let us start with the first example and we will discuss 2 examples in both these cases the players are sort of forgetful of different kind we will see. So if we look at this game so notice that this is a very special game where player 1 is actually playing in both these rounds. So what is happening is that first player 1 picks either an action L1 and R1 then it forgets which action it has picked whether it has picked L1 or R1. So therefore this information set arises that is one at this second level does not remember whether it has played L1 or R1. So therefore in this information set its action set will be the same L1, L2 and R2 and therefore if you want to represent this in the form of a mixed strategy of course you will have to keep all these numbers here L1, L2, R1, R2, R1, L2 and R1, R2 as we have done before. However for behavioral strategies you just need 2 numbers so B, B1, L1 and B2, L2 the other one will be just 1 minus that. Now the mixed strategy seems to have more control over this profiles so for instance if we look at the case where sigma 1, L1, R2 so we can actually put the probability must directly on to this outcome so L1, R2 so this outcome here so at this node we can put a probability mass of 0. Similarly for R1, L2 this outcome also we can actually put a probability mass of 0. Now is this possible to do in the case of behavioral strategy? So what does it mean? So what we are doing is we are putting probability 0 on these 2 numbers this and this and we are putting some positive numbers for these 2 things. So which automatically means that you will have to put some positive masses so if you have to find an equivalent behavioral strategy you will have to put some positive masses on both R1 and L1 you cannot do it otherwise because both L1, L2 and R1, R2 are positive. But at the same time because L1, L2 is also positive so you will have to put positive masses here as well as because R1, R2 is positive you will also have to put positive masses here. So the very fact that you have put positive masses on this 4 edges here the behavioral strategy will also put positive masses on these numbers. You cannot really ensure that these 2 numbers become 0 while here you will get positive masses and that is the difference. So we cannot really represent this kind of a mixed strategy in the form of a behavioral strategy. There is no equivalent behavioral strategy for this mixed strategy. Okay so that is the first example mixed strategy with no equivalent behavioral strategies. The second example is a forgetfulness of a different kind. So here player 1 is again playing first and also playing in the second round. But the point is these 2 nodes in the game tree are connected in an information set which means that player 1 has actually forgotten whether it has played at all or not. So that is a simple way of saying this. So it has played and then at this point if it is unsure whether it has played or not then only it can be connected with a dotted line. So therefore it cannot distinguish between these 2 things. Now here what can happen? So if you are talking about behavioral strategies then it is defined on every node in the information set. So earlier we haven't made it so fine-grained because we never had this kind of examples. So we have only defined the behavioral strategies at a specific information set. But the behavioral strategy can be defined at every node of this game tree. Even if 2 different nodes in this game tree are at the same information set. So we can define the, so what player 1 will do under a behavioral strategy is at this node it will toss a coin and pick L or R which will be given by the corresponding probability masses of this behavioral strategy. Similarly when it reaches this it tosses the same coin because it is the same behavioral strategy it is under the same information set and it goes either L or R based on the outcome of that coin toss. Now what you can observe is that under that behavioral strategy it is possible. So let's say it is half and half. So you can just toss an unbiased coin and that gives half and half. Then in the second round also you are tossing the same coin with probability half and half. So there is a possibility that you will take this path L and R with probability one fourth. But there is no such mixed strategy because mixed strategies are defined based on the PO strategies and what are the PO strategies that are available to player 1 in this case? Only L and R. There is nothing like L, R or L, L in this case. So in this particular case the behavioral strategy is giving you more flexibility than the mixed strategy. So mixed strategy in the mixed strategy world choosing some probability on L means you are going all the way here and choosing R at this node means that you are reaching here. There is no way you can reach this under the under a PO strategy and therefore you cannot reach there with you can reach only this node with zero probability under the mixed strategy. So that is the example where behavioral strategy does not have an equivalent mixed strategy. Think about it. So both these examples are forgetfulness of a different kind. First example the player remembers that it has made a move but forgets which move it has made. In the second example it has actually forgotten whether it has made a move or not. So that is two different kinds of forgetfulness. So this equivalence as we have seen does not hold if the players are forgetful. Now let us dissect the second example a little more so that we can actually save some results about when the behavioral strategy does not have any equivalent mixed strategy. So if we look at the previous example that we have an if we have a specific node let's say non-root node and we are taking the action. So let's say we have a specific node here and we are looking at the path that it takes let's say this particular example may have some other nodes up there. So there is some root above and you are just looking at the path from that root to a specific node here and it is passing through the same information set twice and in the way from its unique path to from the root to this node let's say this is this is my destination node and this path is coming from the root to here. There are two different nodes let's say x and x1 where it is actually intersecting with the same information set and the actions that have been taken in these two different nodes are different. So at this x you are taking the action l and at x1 you are taking the action r. If this happens then you can never ensure an equivalent behavioral equivalent mixed strategy for behavioral strategy on this kind of games. So essentially it is sort of formalizing the example that we have we have just seen so you can formally state in the form of a lemma. So there if there exist a path from the root to some vertex x that passes through the same information set at least twice and if the action leading to x is not the same at each of those vertices then the player of the information set has a behavioral strategy that has no equivalent mixed strategy. So the lemma helps us in proving the following characterization result of equivalence and I am not going to prove this here you can read it. So this is theorem 6.11 of the Maschler's book. So consider an imperfect information extensive form game such that every vertex has at least two actions. Every behavioral strategy has an equivalent mixed strategy if and only if each information set of a player intersects every path emanating from the root at most once. So it is essentially ruling out the situation that we have just seen. If that does not happen then you can always ensure that there exist an equivalent mixed strategy and a behavioral strategy. So you have an equivalent mixed strategy for a behavioral strategy. This is a necessary and sufficient condition for that to happen.