 Now there are two ways by which you can think of randomization. Now one way is just like there are two ways of which you can think of any dynamic game. One is through the normal form, the other is through the extensive form. By in the normal form we just list out the strategies and then think of the game as a static game in those strategies. And in the extensive form we actually observe what the you know the flow of the information that is that is happening within the game. Now randomization can also be thought of in these two different ways. The first way is that well you convert the game to an equivalent normal form and then now in that normal form we have one way of randomizing already which is that a player can pick a pure strategy from that normal form at random. This is what is called a mixed strategy because that is what we refer to as mixed strategy in our in static games. So first way of randomizing is choose a pure strategy at random. Now pure strategies in a dynamic game are functions. So you have a space of functions, set of functions and you choose one of them at random. The probability distribution or the probability with which you choose the every function is what what you are is now the strategic decision. So choose a pure strategy at random. So this is what is called a mixed strategy for a dynamic game. Can you tell me what would be the other way of randomizing? Yeah, so each player would choose a pure strategy at random. So what would be another way by which a player could randomize? So another way of randomization is the following that whenever he gets to an information set the player is aware of the actions that are available at that information set he can choose an action at random at each information set. So at each information set he chooses an action at random. So he chooses a probability distribution on the set of actions available at that information set. So choose an action at random at each information set. This is what is referred to as a behavioral strategy. So the first one chooses a function at random. So it specifies an action at each information set. So at every information, so one of these functions is chosen at random. In the second case we define, so in the first case what we need to do is define one probability distribution which is over the set of all functions or over set of all strategies. In the second case we need to define a probability distribution for each information set but it is over the set of all actions available at that information set. As you can see these are both valid ways of randomizing. A player may want to wait for an information set and randomly choose an action that is the behavioral strategy or he may say well here is my set of strategies I am going to pick one of them at random. Now the second one which is the behavioral strategy is something akin to a delayed commitment. Essentially the player is waiting for the information to be available before he commits to a strategy because the information, the probability with which he is going to take an action depends on the information set he is at whereas the first one is akin to a prior commitment, it is roughly akin to a prior commitment because he is picking a pure strategy at random without waiting for what, without waiting for the intra-gay information to actually get realized, is this clear? Now the question for you is, is there a difference between the two or is there are these equivalent or is one the subset of the other or one more somehow richer than the other? You can see both are very reasonable ways of randomizing so if one is a subset of the other that is an important result which basically it would tell us that well you can focus on one of these classes only, no this is multi-acting. In fact, in a single act, in a single act game this point actually becomes moot. The reason is because along every path a player will play it exactly once let us say and each such way of playing along that path defines for him a pure strategy and that is equivalent to then picking the action in that information set. So these actually become equivalent, I will show this formally but in fact the point is moot in the case of a single act game. The point becomes interesting in a multi-act game because now player has to play multiple times so should he commit to a distribution on the set of pure strategies at the outset or should he do a delayed commitment and play actions at random at each information set, yeah it is a strategy so it is a function that tells you what to do at each information set. That strategy is chosen at random it says that you know play this action at this information set that action at this etc. So based on the information sets definite set of actions are specified so for each information set an action is specified but the strategy as a whole is chosen at random amongst all strategies so related to this question so you can see both of these are basically producing actions at random right but the mechanism for producing the actions is different. In one case you are defining a set of strategies which is functions from information sets to actions which is telling you what action you will be each strategy tells you what action you will be taking at each information set and you pick one such combination at random right that is the mixed strategy. In case so effectively what you are doing is picking actions at random but through these combinations you are picking a particular pure strategy you are picking amongst the set of pure strategies whereas in the second case you are being more explicit you are choosing actions at random at each information set right. So somehow the equivalence of these two has something to do with whether these two in fact lead to similar or classes of distributions on your actions because if they did then you know one way of randomizing would be equivalent to another way of randomizing right. So let me show you an example it is actually true that these are not equivalent in general okay and here is a simple example to show this so let us start so consider this game this is okay so here is player 1 starting here let us say he has two actions L1 R1 then if he plays R1 then player 2 plays here L2 or R2 and player 1 now has to play again okay but player 1 does not get to observe what player 2 has played out here okay so these two nodes here are in the same information set for player 1. The game can end in 4 sorry 5 possible leaf nodes let us just name these leaf nodes and let us see what can happen in each case okay so let us say let us call this outcome O1 this outcome is O2 O3 O4 O5 I am deliberately not putting numbers here because I do not want you to get distracted by the numbers so we have to just think of which outcome is getting realized okay so that then we focus on that rather than you know saying how much payoff each player is getting yes it is a multi-act game player 1 is acting twice no no no it is not in feedback form it is a multi-act game it is not in feedback form yeah that is a different that is what I said last time right a feedback stage wise feedback form is a very specific form of multi-act game multi-act is in general just not single act yeah okay any other question clear okay yeah I mean you could you could but you could potentially reduce it to that sort of form but I mean that is that is not the question here I mean it is whether this can be you could potentially add some fictitious action for player 2 in this stage and then make it into feedback okay but anyway that is not our concern here so if there are five possible outcomes okay and let us what we will do is we will do we will allow players to randomize in these two different ways alright and let us see if we can we actually get some kind of parity between the two ways of randomizing okay so first to begin with let us let us look at let us write out the set of pure strategies for player 1 okay so I think I should write something here also for player 1 so this this is again L1 L1 L1 R1 okay yeah so let us write out what are the pure strategies for player 1 for player 1 has two information sets the node that he starts with and the second one is the this information set and at each information set he has two actions okay so so there are four possible pure strategies therefore L1 here L1 here L1 here R1 here R1 here L1 here and R1 here R1 here okay so I will just write these pure strategies for player 1 I will just write them in short as L L1 L1 L1 L1 R1 R1 L1 and R1 R1 okay how many pure strategies for player 2 he has just one information set he has only two pure strategies which is to play L2 or R2 so what is now a mixed strategy for player 1 a mixed strategy for player 1 is a probability distribution on this set here on this set right which means he is going to play one of these at random okay so mixed set of mixed strategies so let us call this gamma 1 and let us call this gamma 2 set of mixed strategies what are the set of mixed strategies for player 1 it is he has four pure strategies so you can think of it as a let us say y in R4 such that y is greater than equal to 0 and it is a probability distribution right so it is a this probability distribution on this set of pure strategies which has four elements clear so it is specified by so this set is every point in this set is has four coordinates okay out of which one is redundant because this one transpose y is equal to 1 so you can say well this is you give me this is specified by three parameters so mixed strategy is specified by three parameters okay what about the set of behavioral strategies for player 1 so how do I define the set of behavioral strategies for player 1 so he in a behavioral strategy he has to choose an action at random at each information set okay so there are how many information sets to for player 1 so at this information set he chooses an action at random okay so what are so he has to pick a probability distribution on this action set so behavioral strategy will be specified by those by that probability by the probability distributions for each information set right so at this information set he chooses an action at random similarly at this info sorry at this information set he again chooses an action at random okay so a behavioral strategy then would be say he plays problem l1 with probability alpha and r1 with probability 1 minus alpha at this information set and at the other information set he plays l1 with probability beta and r1 with probability 1 minus beta this is a behavioral strategy right so every choice of alpha and beta between 0 and 1 will give me a behavioral strategy is this clear so do I need to say specifically what is the probability of playing this l1 and this r1 here no it is the same because he has they will they will also be beta and 1 minus beta because at this information set he has to play this as a function of just that information right okay so the set of behavioral strategy is therefore is specified by just two parameters alpha, beta now this is immediately telling you something right the set of mixed strategies is specified by three parameters right and the set of behavioral strategies is specified by two parameters. So we cannot really expect any kind of subset type relation like that one is a subset of the other or something like that because these are these are now objects in two different spaces okay these are fundamentally I mean there are more free variables in mixed strategies than there are in behavioral strategies. So we cannot expect a subset type relation but we can so therefore we have to talk of sort of a relation between these that is beyond just inclusion you know that one is a special case of the other because these are this is a this is an R2 these are two it is a distribution yeah no no no so the see it is not a subset you what what you are saying is that I can try to create a relation between the two okay so that is in fact the point that if you are looking for an inclusion type of thing where one is a special case of the other then we have to ask special case and under what notion right special case under what notion right because it is certainly not a subset right then in what sense is this a special case says something has to be preserved under both type of strategies for you to get one as a special case of the other right. So this basically becomes the essential dilemma now if you think about it we have logically said that we could allow players to randomize in these two ways which are which seem both you know reasonable and it should be allowable that players be allowed to randomize in this way but one kind of randomization gives a player who seems to be somehow not you know the same as the other so then we have then we we put we are we get into this position where we now have to choose this is something which we would not want to do as game theorist that we would not want to tie a player's hands and say that this is what he should do we should allow strategic freedom to the player now have we do is there something that one one can now still say that these are in some way similar or equivalent or interchangeable or whatever right. So this is essentially the the juncture we reach and now in so we have to come up with a way by which we say well well is is in what sense are these two strategies you know comparable is clear so yeah yeah so that is part of the that is beginning you are now beginning to think in that direction essentially we have to think of what we what are we meaning by equivalent right so so then you ask ok well equivalent ok well if it is equivalent from the point of view of payoffs right then we can ask ok well if one framework you can say as well I am asking for equivalence in terms of payoffs which means that the problem which means that the payoff that I get the expected payoff that I get under one I should also be able to get it under the other by a suitable choice means for all mixed strategies that give me a certain expected payoff can I produce a behavioral strategy that gives me a gives me the same payoff this could be one's notion of equivalence because then in that case I do not care about whether I use and as a as a player I do not care and likewise as a as an observer of the game I do not care what the player is doing whether he is randomizing this way or randomizing that way but you can see there is one catch there as well because the payoff the expected payoff depends not just on what this player is doing but also what other players are doing so then we have to define expected in a suit expected payoff as being the same in a certain in a very strong sense that it means that regardless of what others do this guy should be able to simulate the same same payoff with two different types of randomizations right yeah on join distribution and marginal distribution on roughly yes yeah correct so you are absolutely right essentially these two ways of randomizing they define for you a joint a joint distribution on the probable on the set of nodes of the directory they tell you with what probability if I tell you a make strategy or a behavioral strategy for each player ok whatever combination some may play mix some may play behavioral effectively it from there I should be able to compute the probability of the appearance of the probability that the game history passes through each node for every node I should be able to compute such a probability right so you are absolutely right but then now I can compute that but then now for equivalence what should I ask for so what Ashwin was saying was that you can ask for something you can you start with the payoff let us just say ok let the play let us just ask that the probability of reaching the leaf nodes is the same right but then we have a joint distribution which tells us the probability of reaching any node not just the reef node right so then that gives you that that then demands us maybe maybe should we be extending that why leaf node only should we be allowing for all nodes if we allow for all nodes does it become too restrictive etc etc these are all questions are involved in in in trying to come up with the right notion of equivalence right so so the this this question actually was is is actually very interesting and it is it turns out it is related finally whether you can whether there is any equivalence between these two has something finally to do with the amount with memory these this issue comes up only in multi-act games and as I said the fundamental new feature from in multi-act games as opposed to single-act games is the issue of memory what does a player know of what he knew earlier ok and whether these two forms of randomization this form of randomization or the other mixed and behavioral whether these two can be interchanged in a in a in an effective in a you know in a in an effective way is eventually decided by what a player can remember ok so the memory has memory has a huge role to play here ok so so what I will tell you is this notion of equivalence so so the way so it turns out actually that if you restrict ourselves to only leaf nodes right here if you restrict ourselves to only leaf nodes it becomes to it becomes to let us one is it does not become elegant the other is it also leads to a whole bunch of other loopholes which we which we will have to plug later ok so there are much a much better notion of equivalence is is that you ask for equivalent that you can simulate the same distribution for every node in the ground for every node in the tree ok what we will do is we have now two different ways of randomizing so we will let sigma i be mixed strategy of player i so this is a probability distribution on so probability distribution on gamma i which is a set of pure strategy and bi is the behavioral strategy is a behavioral strategy so behavioral strategy is a conditional probability distribution now given each information set it produces a probability distribution on the set of actions so it is not it is a so it is actually a collection of conditional probability distributions ok so this is so the way we write this is that bi of this given an information set eta i is a probability distribution on actions available at eta i ok and I will put in another notation here mu i this will stand for any either of the above so when I write for all mu i for example it would mean that it could be it is essentially all mixed and all behavior ok that is what it would mean ok now so here is the definition of equivalence mixed strategy sigma i a mixed strategy a mixed strategy sigma i and a behavioral strategy bi are equivalent are said to be equivalent if for every mixed stroke behavioral strategy mu i behavioral strategy mu minus i sorry strategy mu minus i and every vertex x in the game tree we have the following probability of reaching the vertex under sigma i when player i plays sigma i and others play mu minus i is equal to the probability of reaching that vertex x when player i plays b behavioral strategy bi and others play mu minus i so notice how strong this is but this is how this is what we need otherwise we will end up having too many loopholes to fill as I said ok so this is strong why is this so we are saying that two notions two stripes of strategies sigma i is equivalent to bi if for everything that the others play they could play mixed stroke or they could be behavioral that is and that is encompassed into mu minus i so any mixed or behavioral strategy combination or that the others could play called mu minus i and for every vertex in the tree these probabilities are the same ok this is what is this probability this is the probability that you reach vertex x when player i plays sigma i and others have playing mu minus i and this is the probability that of reaching vertex x when player i plays bi and others play mu minus i this is clear so once I specify a profile of strategies combination of mixed behavioral whatever it produces for me a probability distribution on the nodes of the tree so these two quantities are well defined what I am asking for is that regardless of what the others play I should be able to get the same distribution that I get from sigma i that I should be able to get the same one from bi ok so this means that this makes these two strategies equivalent ok so the I switch from sigma i to bi I should be able to if I if the distribution of probability distribution on the nodes remains the same and others and it remains the same regardless of what the others play ok then regardless of remains I mean the equality holds regardless of what the others play ok of course the distribution will keep changing with the choice of mu minus i but but when it changes it changes to the same the the one that I get with sigma i is equal to the one that I get with bi is this clear so now as a result of this we get the following simple fact that if if sigma i is is equivalent to bi is if the mixed strategy sigma is equivalent to a behavioral strategy bi then for all mu minus i the the payoff that a player or the the cause that a player gets when he plays sigma i and others play mu minus i is equal to the cause that he would get when he plays bi and others play mu minus i is clear of course the value of the cost will will depend on what others are playing but regardless of what others play I can get the same values by switching from bi to sigma sigma i to bi is this clear ok right so and now this is equivalence between strategies of one player if you have this way equivalence between the strategies of all players right then you would have that for this is true for every player ok every player will be able to switch from between behavioral and mixed