 In the context of extensive form games, we have so far discussed about pure strategies. We haven't spoken about how we can mix the strategies or somewhat probabilistically pick these strategies. So unlike the normal form games where there was only one kind of randomization available, which is like you have the set of pure strategies and you are randomizing picking any of those strategies in a probabilistic manner, which we call the mixed strategy. In extensive form games, because the extensive form games have multiple stages, you can think about the randomization in two different ways. One is where you are enumerating all possible pure strategies, that is the complete plan at every stage of the game and randomize then assign probabilities to those strategies. Or you can think about an independent draw of a probabilistic draw at every information set of this game. So why didn't we talk about this randomization in the case of perfect information extensive form games? Because we already knew that it is possible to find a pure strategy equilibrium, which was the sub game perfect Nash equilibrium, which also happens to be the pure strategy Nash equilibrium. Because PIEFG is one such kind of game where the pure strategy Nash equilibrium is guaranteed to exist and that happens to be the sub game perfect Nash equilibrium for finite PIEFGs, then we never bothered about going for randomized strategies. But as we said here, in the case of imperfect information extensive form games, we will have to go for randomized strategies and that randomization can happen in two different ways. So let us look at directly the example and try to see what is the difference between these two types of randomization. So let us look at this example on the left hand side. So here we have two players, player one is playing at the root and also at this information set, which is at the third level. And in between player two makes another move. So we are going to just denote the first information set which is a single done for player one with I11 and the second information set as I12. Similarly player two has only one information set which can be represented as I21. Now what is the set of all possible strategies for pure strategies for player one? We have done this earlier. So because in a specific information set for all the nodes that is living in the same information set should have the same set of strategies. So therefore in both these nodes here, the strategies are the same L2R2 in both these cases. So therefore player one can have four possible pure strategies. This is the set of all possible pure actions, pure strategies that player one can pick. So let us focus only on player one because for player two it's only two of them and there is no difference between the two type of randomization that we are talking about. So now by the definition of mixed strategy in the classical way that the way we have defined it for the normal form games, we are going to assign probabilities for each of these pure strategies. So therefore a mixed strategy in this context for player one will be a probability mass which gives probabilities to all these pure strategies. And these numbers should always be non-negative and should sum to one. Now you can think of another kind of randomization which does not list out all possible strategies of this player at the very beginning. So you can think of that why do you need to list out all the possible strategies? Player one can just toss a coin at this information set and decide whether to play L1 or R1. Similarly, when it reaches this information set I12, it can do a very similar thing. It can toss another coin or maybe the same coin which will give it a different probability distribution over the strategies L2 and R2. That looks more natural in this case because there is this amount of independence in two different information sets. So the type of randomization that we just discussed is what we are going to call. So those kind of probabilistic strategies we are going to call them behavioral strategies. So this is different from the mixed strategy definition that we have discussed so far. And in this and the following modules we will see that there is a certain amount of difference. You cannot really transform one strategy into another and vice versa all the time. So what, how can we formally define a behavioral strategy? So let's say B1, I11. So this is the behavioral strategy at the first information set for player one. And that is a probability distribution over these two strategies so L1 and R1. Normally the behavioral strategy at the second information set I12 is a probability distribution over this L2 and R2. So this is just formally capturing that stuff, that type of randomization that we just discussed. So formally the behavioral strategy of a player in an IIEFG is a function that maps each of her information sets to a probability distribution over the set of possible actions in that information set. So the difference between this behavioral strategy and the traditional mixed strategy is that we are going one information set at a time and we are randomizing at an information set level. We will definitely talk about the relationship between mixed and behavioral strategies. One observation that we can make from this example is that the mixed strategies here are living in a higher dimensional space. So it is in R to the 4 because there are 4 pure strategies and you are putting probability masses on them while the behavioral strategies are living in 2 separate 2 dimensional spaces. So that way it seems like mixed strategies are a little richer, larger concept but we will see that that's not completely true for all sorts of games. So we can ask questions like can a player attain a higher payoff in one strategy than the other and the most important question is can we have an equivalence. Which kind of games we can have equivalence so that we really don't need to think about whether it is a mixed strategy or a behavioral strategy. It will be the same, we can use them interchangeably. So to talk about equivalence we will have to first define how we are going to define the equivalence between 2 different kinds of strategies. And in order to do that we can define the equivalence in terms of the probability of reaching a particular vertex or a history. Let's say x. So suppose in this example, in the same example we have a specific node x and we are asking what is the probability of reaching this particular node under a behavioral strategy, this or a mixed strategy, this. So we can start with the mixed strategy. So what is the probability that we will reach this vertex x. So this is going to be, so definitely we will have to, for this to happen we will have to look at what is the probability with which you pick this r1, this strategy here so that you reach here. And then what is the probability that player 2 is picking the action r. And therefore we can actually write it. So sigma 1 r1 times sigma 2 r2 because these two players are picking their strategies independently. Players in this case will always pick strategies independently. So we can expand this. Remember this sigma 1 was essentially a function of two random variables. The first random variable was taking values of the action that is happening in the first round. And the second random variable was taking the values in the second round or second round of player 1 which is the third round of this game. Now if we just ask what is the probability that sigma 1 l1 then we will have to marginalize over sigma 1. So we will have to sum sigma 1 l1 l2 and sigma 1 l1 r2. So similarly for finding the r1 we will have to do this exercise. So sigma 1 r1 l2 plus sigma 1 r1 r2. So that is written here and then multiply with sigma 2 r. Now what happens when we look at the behavioral strategy? The probability that the game reaches the same node, same node x under this behavioral strategy is given by b1 i1 1. So the behavioral strategy is rather easier to capture because you are just looking at the behavioral strategy at this information set and what is the value? So what is this probability when you are picking r1? And then you look at the behavioral strategy of the other player 2 when it is picking r. So this is the two different ways you can reach the same node and these are the probabilities. And when we are talking about equivalence very naturally these two numbers should equate. They should be the same. So let me just go over and define that a little formally. The different players can choose different kinds of strategies. So this is perfectly fine. So even if player 1, so if you are talking about a specific player let's say player 1, it does not really matter whether the other player is picking a mixed strategy or a behavioral strategy. So if player 1 chooses the sigma 1 above and the other player is player 2 is choosing the behavioral strategy then you can write this in a combination of this. So the first one is coming from the mixed strategy of this player but the second one is coming from the behavioral strategy of the second player. So we will define the equivalence between a mixed strategy and a behavioral strategy in the following way. So a mixed strategy is sigma i and a behavioral strategy bi of a player in an IIEFG are equivalent if for every mixed or behavioral strategy vector xi minus i, so that means it is holding for all possible strategy profiles of the other players and at every vertex in that game tree this equivalence. So that is the other players are playing their favorite strategy but they are holding on to that strategy. This could be a mixed strategy or a behavioral strategy or different players might pick different kinds of mixed and behavioral strategy but despite that the probability of reaching this node under the mixed strategy for player i should be the same as under that behavioral strategy for the same player. So if we look at the example above then we can actually work this numbers out. So we can find that what is the behavioral strategy. So the behavioral probability mass on L1, so this is the first outcome. So if you look at the first node here, so this is under that behavioral strategy b1 of I11 of L1 this has to be the same as the probability under that behavioral strategy under the mixed strategy that is coming L1 which is nothing but the marginal the sum of the marginals. Similarly, you can do the same exercise for R1 what is the probability of reaching that node so this one was the first node the second one is about this node and we are equating the behavioral strategies there and then we look at the following the final nodes. So what is the probability that if it takes this L2 so and go to the terminal nodes then what is the what are the corresponding probabilities. You can just do the very similar exercise by marginalizing appropriately all these numbers. So this I have written as a short hand notation what actually happens is you have to so here we will have an expression like R1 that is the first round player 1 is playing R1 and in the second round it is playing L2 and you are going to divide it with the sum of this marginal so R1 and L1 plus sigma 1 of R1 times R2 so this will be so this will the denominator in that case will be nothing but sigma 1 of R1 and that is the reason you can actually write it as a conditional probability you know that this this divided by this number is nothing but L2 given R1. So you can do this exercise yourself and similarly for the other one so this is going to be the number so if these two strategies are to be same then this numbers should be should be equal then only we can call that B1 and sigma 1 are equivalent according to our definition. So let us make one very important observation and there is a claim that is going to come so whenever we are talking about this equivalence in this example we have looked at the equivalence at every stage but it is not really necessary to look at the equivalence at every stage of this game rather this claim says that it is sufficient to check the equivalence only at the lift nodes and you can prove this claim formally the the intuition is the following how do you find the the equivalence at the probability of reaching a non-terminal node non-lift node is by looking by summing the corresponding probabilities in its subtree if you restrict in this game tree to that specific node where you that intermediate node where you want to find out the probability of reaching that node you are essentially going to submit over all the the probabilities at all its leaf nodes in its subtree so therefore if we have the equivalence at the lift nodes you are guaranteed to have the equivalence at those nodes so that essentially shows that it is not really required to look at the the intermediate nodes it's just redundant you can only ensure the equivalence at the lift nodes and that be sufficient to make sure that these two strategies mixed and behavioral strategies are equivalent and this argument can actually be extended further even for the utility description so utility equivalence says that if you have two strategies mixed and behavioral strategies which are equivalent then for every mixed or behavioral strategy vector of the other players which is given by xi minus xi the the equivalent also holds for the utilities so that is that is the utility equivalence result this comes as a byproduct of the equivalence of the of the behavioral and the mixed strategies now you can repeat this argument for any equivalent mixed and behavioral strategy profiles that is it is for all the players if they are equivalent so here we are going to call this sigma and b are equivalent that means for every player the corresponding sigma i's and bi's are equivalent then you have the utility at at that strategy profile mixed strategy profile and the behavioral strategy profile to be also the same this is very straightforward to show