 So, an extensive form is a tree, a specific vertex indicating this starting node, the starting point of the game. You have a cost for each player at each leaf node, leaf node or also called terminal node, a partition of the nodes of the tree into player sets. So, you partition the nodes of the tree into player sets. So, really all nodes, so I should be more precise here all nodes except for except the leaf node, except terminal nodes. These are partitioned into what are called player sets, a partition of the player sets to information sets. So, into information sets, the information sets have some requirements such that this same number of branches from each node in the same information set. Now, there is an additional condition which is also often put you will find this in some books. So, let me put that here as us with a star. The additional condition is that no node is descendant of another node in the same information set. So, I will come to why they all I will explain all of these things now. So, an extensive form is defined in this way. So, you have a tree, you take a particular node of the tree and that becomes your starting node. So, the game starts from that node. The nodes of the tree, the leaf nodes are where you list out the payoffs. So, what the game ends when you reach a leaf node. After that, at the leaf node, nobody plays, it is just a sort of a placeholder for you to know that the game has ended. So, what we call terminal state and control and so on. So, it is the game ends there and you get your payoffs. There is a two-pole of payoffs listed for all the players at each leaf node. The other nodes of the tree are partitioned into what are called player sets. Player sets tell a player when it is his turn to play. So, when a particular node belongs to player 1's player set, it means that when the game comes to that node, it is player 1's turn to play. So, all the nodes have some of the other, it is a partition. So, which means all the nodes have some of the other player, every node has some of the other player assigned to it as the player whose turn it is to play. And again, it is a partition which means there is no conflict. There cannot be two players whose turn it is to play at that, at the same node. So, at every node, there is a player whose turn it is to play, exactly one player whose turn it is to play. Now, the player set of a particular player is a collection of nodes. This further is subdivided or partitioned into what we call information sets. And the understanding is that if one or more nodes are in the same information set, then the player cannot distinguish between the nodes in that information set. This is clear? So, if you go back here, let us take this structure, the node, this is your extensive form. The player sets are like this. This is the player set for player 1, just this node, single node. The player set for player 2 are these three nodes, this one, this one and this one. These have been further partitioned into two information sets. There is one information set here, this bubble. And there is another information set, trivial one here, which is this, clear? So, and the understanding is as I said, player cannot distinguish between any two nodes that are in the same information set. So, these two nodes for player 2 are in the same information set. So, then for him, these two nodes are actually the same. He has no way of distinguishing what between the two nodes. The extensive form also has this additional constraint that you cannot just arbitrarily subdivide nodes and create information sets. You can decide who plays where arbitrarily, more or less arbitrarily. But you cannot subdivide player sets arbitrarily and create information sets. And the reason for that is that there is this constraint that the same number of branches must come out of each node in the same information set. So, here these two nodes have the same number of branches coming out of them. So, these two nodes can be put in the same information set. If you imagine there was another strategy here, let us say, sorry, another action here for player 2, then these two could not have been in the same information set, that is not allowed. So, yeah. So, this is then the next natural question. So, why is it that, why is this requirement there? Exactly. So, this is basically coming out of the very consequence, out of the fact that player knows what actions are available to him. So, if there are more actions available here than there are here, then he could have distinguished between these two nodes. Then it does not make sense to say that they are in the same information set. You are basically contradicting yourself. You are saying that he cannot distinguish between these two, but he can play, there is a distinct different other action which he can play at the other node. So, what this means is his actions are also a function of his information. The actions which means the actions available to him are a function of his information. So, if he has the same information, then it necessarily means that he has the same set of available actions. If there is another action available in some other scenario, then that means there is some reason for him to distinguish between that scenario and this scenario. 25. Yeah. You are there. Which, where would you add the phantom node? You say you had L2, M2, R2, 1. In, out here. Yeah. One of these. No. Yeah. No. Yeah. Yeah. So, then it could not have been the same. The game. Yeah. Yeah. Because there would be a different set of strategies. It would match, that could happen. That is a different problem. So, this is, then that is, we are not seeing that two games are equivalent if their answers match. That is not the claim, right? So, see, you could have added another phantom action here, for example, but then it would be a different game. And in some cases, now here, of course, this is zero sum, probably it would not matter. But if it is non-zero sum, it could create problems. Point is, you have to have, you need to respect the physics of the, of the situation here. That time is flowing in this particular direction, right? As we start from the root node down towards the leaf node, that the extensive form game keeps track of the exact history of actions that have taken place. Players may not know which node they are on, but when they take an action, it gets registered in the extensive form, right? That, so player 1 plays M1, but player 2 does not know whether it is M1 that has been played or L1. Then player 2 plays, say, L2. The game reaches this node. This is the exact history that has happened. And player 2 could not, the very fact that player 2 could not have distinguished between these two has to be reflected in the fact that he has the same set of actions. Now, here actually, this requirement is stated in a very mild way. It is saying that, you know, the same number of branches, really what you actually are asking for is the same action set, exact same action set. You know, technically, you are not even, shouldn't you, the actions, I have not given labels to the edges here, for example, what the labels are like L2, M2, R2 and so on. That is, I have ignored that part, because I just want you to focus on the structure. The point is that even if there was some label that was different, even he had the two same two actions, same number of actions, but two, but their names were different. So, he could have had the action to go up and down in one node and left and right in another node. But physically, left and right means different, something different from right up and down, then these are two different actions. So, the point is the player should have no reason to distinguish between these. So, the information should be exactly the same. And we have to, you know, whatever the ways by which he could have snooped in and figured out something about where he is, all of that has to be already factored in and there should be no other ways for him to know any, you know, about where he is after this. Is this clear? Now, this additional condition, last condition here is a more complicated condition. Here, what is happening is, let me, I will just tell you this briefly. So, here what it is disallowing is this. Suppose, there is a node here. So, let us say player 2 has to play here at this some node. So, it is part of some larger tree. And then there is another node, which is player 2's, which is also player 2's node. And these two nodes are in the same information set. So, now what this means is player 2 has taken and play this node, the second one, the one below follows from the one above. Means there is a sequence of actions after which this node is arising. Let us take for simplicity that it comes out of the immediate action. It follows from the immediate action. So, player 2 takes some action and then he gets to a next node. Let us call this node x, let us call this node y. He takes an action at node x. It goes, the game goes to node y. But we are saying that node x and node y are in the same information set of player 2. What is the meaning of this? So, it means that he is not even aware if he has taken an action. For him, it is the game, he has taken an action, the game has moved. The game is physically now at a different node. The history of the game has evolved. But he is not aware that he has in fact taken an action. Now, this you can allow or disallow. This condition is a form of absent-mindedness. You have done something but you do not remember that you have done it. Now, whether you allow absent-mindedness or not depends on the proportion of absent-minded people in the world. So, in game theories is rather large. So, we can allow this. I am not fully in favor of removing this. This is only a kind of a technical thing. Whether you want to allow for a player to, so when a player takes an action, is he actually aware that his action has in fact been taken? So, this I will give you an example. For instance, so a place where this is not just, I mean I am giving it, it is a form of absent-mindedness because I will like to give personalities to the player. But you do not have to think of it that way. Think of it like you are in some, you are in a spaceship in outer space. You take an action, it is supposed to spin you in one particular direction or turn you in some direction. The world around you looks exactly the same. You have no way of knowing whether you when after you press the button, whether your spaceship moved or not did not move. You will try to press it again. It is that sort of a situation. Essentially, there is no way for you to know whether the action that you supposedly took has in fact been taken. So, the game next time when maybe you press the button, then the spaceship will actually show you some, your new planet or whatever and you will know that it had worked. So, you have no way of knowing whether the button was not working or you had not taken the action or whatever. The action was not taken or the action was taken and yet the world looks the same. So, this leads to all sorts of almost nearly paradoxical type of situations. So, people tend to avoid it. It also leads to interesting questions. So, one can for their own sake you can include it. So, this is more of a let us say an optional condition as far as whether we allow this in the model or not. Yeah, not necessary. No, no, no, no, no. The reason is because again information. You can very well have situations where it is a, this is player one, let us say player one turn to play after which so player one plays and then again it is player one turn to play and you have these two in the same information set. Player one has played then two different he could have played say left or right and the game reaches one of these two if he plays L he gets to this note if he plays R he gets to this note. Now, what does this model tell me? So, he has played L or R whatever let us say he plays L the game comes here he play or he plays R and the game comes here. But when he comes to these one of these two notes he does not know whether it was whether he has got here due to his because he played L or because he got here because he has played R. He is basically forgotten what he played. He does know that he has played he remembers that he has played that is different from this where he does not even remember if he has in fact played. Is this clear? But he here he remembers that he has played but he does not remember what he had played between these two notes and so it is fine. So, your question was is it can you necessarily combine these two? The answer is no you there is a there is a genuine loss of information here again you will have to do it prove it properly. So, you cannot this there is a loss of information the loss of information is means that you cannot combine these two notes collapse them into one because no so it matters for example the what you would so player a player that plays L or R but cannot record whether he has played L or R and later has to then take an action is different from a player who has to play L and R and directly come to this these are very two different situations. See essentially when loss of information means it is essentially something is not being recorded in memory your chip does not have that memory or your brain does not have that memory whatever something is not being recorded as an event which has actually happened and I mean then to say that that that can in fact be collapsed to a situation to a case where he had that information is that is not true no collapse would work means. So, yeah there are there are you can in fact there are not only that the set of equilibria itself changes when you allow for information where you are with memory and without memory the set of strategies itself is changes and that leads to a different set of equilibria. So, if a player is node is following his own action right. So, player ones next node is following from his own action we cannot it is not necessary that you can combine these two into one sort of collapse them into one action. So, the essentially the game where you take two consecutive actions but you forget that the second instance you forget what you had played first is different from the game where you take one action collectively for as if you know at the collective action for the last step itself. Yeah, yeah of course, of course, of course he does not remember it could but it could matter. No, no, yeah it could it could exactly that is the point it could matter. So, I mean so, so this kind of these sort of things of loss of memory and so on are actually very very important I mean. So, if you if you go and watch some of my lectures in on stochastic control problems right this is essentially that the exact same thing that happens where you have a player taking sequence of actions but then there is a limited amount of memory from one step to the next step and or there is noisy memory you know there is memory but with corrupted with noise it qualitatively changes the problem. Yeah, we will do this and much more many more exotic ones. Okay, so can you tell me what are the how do I express a simultaneous move game as an extensive form. So, what is the simultaneous move how would I characterize the simultaneous move game as an extensive form game, which partition all of player 2's nodes should be in the same information. So, in short each player has only exactly one information set each player has exactly one information set this is a simultaneous move game why is this a simultaneous move game because if at least one player has more than one information set then that means that there is some situation in which that player has two pieces of there are two different two situations in which the player that player has two different pieces of information. If he has if there is only one information set which means regardless of what has happened he he has he really has just one information right. So, then if that is a simultaneous move game. So, each player has exactly one information set so our information set is equal to player set. So, that means at all the nodes that he is playing he is he has the same information and this is true for every player. So, the other extreme is the game of perfect information. So, what is the game of perfect information in which a game of perfect information is one where player knows exactly what which node he is at then what would that be each information set is a singleton right. So, there cannot there is no information set with more than one node in it. So, each information set so there are as many information sets as there are number of nodes in the player set. So, this is the other extreme which is called case of perfect information. Usually all the other games that we had which are so this one for this game for example is a game of I said imperfect information and it is also not a clearly not a simultaneous move game and it is imperfect information because some this second second player does not did not have access to this whether the first player whether he was at this node or this node. Now, the what are the strategies of a player. So, let us write this the set of strategies strategies of player i this is the set of strategies of player i these are. So, let us write let us say i i the set of information sets of player i. So, a strategy is going to be a mapping form an information set to the actions in that information set. So, let us write let us write let us say u i subscript eta i be the set of actions to player i at information set eta i in i and let us write capital U i as simply the union over eta i in i i of u i. So, this is set of all actions of player i. So, then what is the set of strategies of player i can be expressed this in terms of the notation we have just written. So, you have the set of information sets for the set of actions available at each information set. So, this set of strategies then is of is set of functions like this these functions will map each information set to an action each information set is being mapped to an action and then such and you have the constraint that at each information set you have to take an action from those that are available. So, you have to take an action this is the set of set of strategies. Now, as I had mentioned the way the game proceeds is that you start from the starting point of the tree that is in some players information set or player set he plays an action then someone else plays an action and so on and then eventually the game when it reaches a leaf node the game ends. So, at the leaf node you have the realized payoffs for all the player. Now, if I tell you the strategies if I tell you a profile of strategies gamma 1 to gamma n for my n players does this define for you a path through the tree starting from the root node to some leaf node it does because the first acting player takes some action whatever it is the node comes to it is going to be some turn of some player to play his strategy that player strategy will tell you what he is going to do in that information set wherever that node is then that guy then the next guy will play or whatever. So, every profile of strategies this defines a path from root root node to a leaf node or a terminal node. So, which means that each profile of strategies defines for you a payoff for each player. So, the payoff then for player i from the end from the strategies of the n players. So, this is there this defines for you a payoff. So, it is a way so if I give you a profile of strategies which are functions from the information sets to actions that defines for me a payoff for the players and then that define from that profile of that defines a payoff and once I have a payoff I can talk of a Nash equilibrium. So, once again we are in it is the same situation as before you players are picking their gammas which is their functions simultaneously. It is a simultaneous choice of the functions and so we can define the Nash equilibrium in the space of such functions. So, gamma 1 star to gamma n star is a Nash equilibrium if ji of gamma 1 star to gamma n star is less than equal to ji of gamma i comma gamma minus i star for all gamma i in capital gamma i that is it. And you can see this is basically our way of we have essentially generalized the way we were looking for Nash equilibrium by writing a normal form and so on this is what we were doing is effectively just this. This is also generalizing the way we were looking for Nash equilibrium of static games because static games are just a special case where each player has one information set.