 So far we have discussed games which are one shot, that means they can be completed in one specific round. So all the games that we have discussed has this property that whenever the players choose their strategies the outcome happens and that is the end of the game. But we have also seen some examples of games for instance chess where this is not the case. The players do not choose their actions or strategies all at once. They look at the progression of the game. So the game does not end in one round. First one player makes their move and then the second player makes their move and so on. So this to represent games of this kind which are sequential in nature we need a richer representation of the games and that is what we are going to discuss in this module. So this sequence of actions will sometimes call them as the history of the game. The first representation type of this kind of game is called the perfect information extensive form games. So extensive form games generally refers to the case where you are representing the game as a game tree that is it is not just represented in the form of a matrix and it automatically means that there are multiple stages inside this game to represent that we use the term extensive form. The perfect information comes from the fact that all the players can look at the history and look at all the moves of the other players. So in some sense every intermediate state of the game in this extensive form game is completely visible to all the players. So this is true for games like chess where you can see the board position and whatever move the other player has made. But it is not true for games for instance like cards where you can see what your hand is and you can see what cards have been played on the table. But you cannot see the hands of the other players. You may have a guess but you still have certain amount of uncertainty what is the exact state of the game. So those kind of games we are going to discuss later. So first is the perfect information game where every move and every state of the game is visible to all the players. So because this is a richer representation expect a little more notation. We will develop all those things and try to explain that as detailed as possible. So let us look at the example of a brother sister chocolate division game. So suppose a mother wants to divide two chocolates to indivisible chocolates. Let us assume that each chocolate cannot be divided into further pieces. And the mother gives this chocolate to the elder brother and asks him to divide it in a way such that the sister can be happy with it. Both of them can be happy with it. So the brother has these choices. Either it can pick both two chocolates to himself and give nothing to the sister. It can divide it one one or it can give both the chocolates to the sister. And sister can by looking at this division it has two choices. Either it can accept that division or it can reject that division. And now the mother has said the rule that if the sister is unhappy and rejects this division offer then both of you get nothing. So I will take away both the chocolates and you won't get anything. So this is the setting of the game in words. Now we are going to represent this as an extensive form. How should we do it? So we can draw a game tree as we did in the case of Chase. So at the first note the brother is the player and all the subsequent nodes subsequent intermediate nodes in this case is the sister is the player. And this action is essentially whatever the brother has divided. So two zero means it has given both of it has kept two for himself and zero for the sister. One one means it's equally divided zero two as we it's evident what it means. Now the sister at each of these stages can either say accept or reject. If it accepts the corresponding division is is the final utility of both these players. And if it rejects in all these cases they both of them get zero because both the chocolates are taken away. So how should we formally capture this? So like normal form games we will also represent the perfect information extensive form game using a tuple. But now the tuples will have some more numbers. We still have the set of players state of actions and set of utilities but there are certain additional things. So let's look at this one by one. So first is the script H which is the sequence of all actions or histories sequence of actions rather. So it's not just one action for the player it's the sequence of action of all the all the players. If you look at a specific history so let's say if this is the history then all the sequence of actions that led to this path in the game tree is considered the history of that game. Similarly if you pick any intermediate node then this part is the is the history. So it satisfies certain properties the first thing is the null the empty action must already be in this set script H. That means the the origin node should be inside this history. The second condition is that if we have some history which is in this history set H then every subsequence of H starting from the root should also be a member of a script of H. So this means that if you are if you are looking at this kind of a history so that there are two actions so action one and action two then any history. So here there exists only one sub history which looks at just this action that should also be in the history set and we are going to name a specific type of history to be a terminal history. So this type of histories are for terminal history. So what what it is so it is a sequence of actions. Let's say it has passed already t rounds. So in the first round the 0th round the player has some player has played a 0 in the first round the next player has played a 1 and so on until t minus one the action t minus a t minus one was played. And you cannot find any other feasible action a t in that set in that action set such that if you append that action at the end of this previous history that will remain in H. So in other words it is already reached a terminal node so there does not exist any any other action that you can append and still be inside that feasible sequence of actions set. So if that happens that kind of a history we call the terminal history. So in order to distinguish those nodes or those histories so we are making a specific name for that sub history. So all the histories where you actually reach the terminal node we are going to call the set of all terminal histories and this is represented by Z. The next function that we are going to define is script text as shown in this in this couple here. So what does this function do this is nothing but the action selection function. So its purpose is to find out at a non terminal history what is the feasible set of actions available to a player. So because at the end at the terminal history there should not be any action left because there is no player who will play at that that is the end of the game. So for all the non terminal nodes you so if you apply this function X it will give you the set of actions that is available to that non terminal node. So for instance if you apply this function X to a history non terminal history H then it will give you a set let us say A0, A1, A2 which is a feasible set of actions that you can play at that level. And now the next thing is who will play those actions so that is given by this function which is the player function. So this is the next entry in this couple it is telling you that who is going to play there. So again the non terminal histories and it is mapping to the set N. So this is giving you the player who will play and it has this feasible actions. And finally we have the utility for each of these players and the utilities are going to be defined only at the terminal nodes because that's the end of the game and that is when the utilities are realized. For every non terminal node there is no utility associated. So now it is the time to talk about the strategy of a player. We have informally said when we discuss the game of chess as a mapping from the state of the game to the set of actions but now we can use the notation that we have already discussed. So what is this strategy set? So first we look at a specific history. So let's say the history is H and also we know that this particular player we are focusing on player I. That player is a player and that's at that history. So we look at all possible such histories where player I is going to play and take the Cartesian product of the corresponding action sets. So in the previous case if you look at this example, so let's say we are looking at the second player, let's say sister. So sister let's say is the second player and brother is the first player. Then sister is playing at three non terminal nodes here, here and here. So therefore the sister's strategies will be defined by a couple of three things as we have defined here as this shows. So it will be the let's say the first history is H1, so that was 20. The second history was 11 and these are nothing but the actions that has been taken so far. So 20 is one action that is one history, 11 and 02 is the third history. So if you look at the H3 which is 02 then you know that these are the three places where player II is going to play and therefore there will be a couple of three things in its strategy set. So S2 will be the elements which is like let's say accept, accept, accept a couple of three things. Similarly accept, accept, reject is another history, another strategy for player II and so on you can fill it up with all possible such cases. So let us do a kind of a quick review of all these parts, all the components of this perfect information extensive form game by looking at the same example that we have started. So what is the set of players in this case? So we are representing the same game, the perfect information extensive form game using the notation that we have done. So there are two players, player I and player II which is the brother and the sister respectively. Now the action sets are actions, possible actions are this 20, 11, 01 and accept or reject. So this is essentially taking the union of all possible actions that is available inside this game. Now you know the histories, of course the null history is the 0th history from which it starts. Then you have the histories of 20 that is the first level of this game. So this is one history, so this history is being represented by 20, similarly 11 and 02. And then you have this terminal histories which are 20, a. So once you take that you reach in this particular terminal history similarly 20, r and it is just filling up the rest of the things. Now we are just taking out all these terminal histories, all these nodes at the end and calling that as the set, the set of terminal histories. Now what are the action sets that are action available for the players at each of these histories? So x of phi which is the root node where this player is playing. So it has three actions, 20, 11 and 02 and at every history, so let's say at the history 20, the sets in this case are equal. So both all of this x, 20, 11 and 02 are all a, r because the second player is taking the same set of actions. It has the same set of actions at every second round of this game. Now what do we know is the player, the first player is playing at the 0th node, the root node and at every other non-terminal history player 2 is going to play. That is represented by this function here. Now what do we have? We have these utilities, 20, a. This utility at the terminal node is given by 2, the utility of player when the terminal histories 1, 1 and a. So this is the utility for player 1 and for all the other terminal histories, all the other terminal histories in this case, the utility is 0, you can verify that. Similarly for player 2, at these two terminal histories, the utility is positive and that has been listed here, all the other utilities are 0 for that player. So now the interesting part is that what is the strategy of player 1? So the strategy, so this player is playing only at the non-terminal history of phi, so that is the root node and it has three actions available to it. So that is given by this one and that is going to be the strategy set for player 1. So that means if it is at 0, which of these things it can pick is given by s1. While for player 2 as we have just explained, it's a Cartesian product of three sets because it is a player at three different non-terminal histories and therefore it will be given giving this kind of a strategy set for this second player. Now what we can do, so this is the representation of this game, the perfect information extensive form game. Now we will always try to connect between different notions of representing the game. It is more convenient to express this game, the game that we have just discussed in an extensive form but it is not impossible to represent the same game in a normal form. So in fact we can do that and we will see that this perfect information extensive form game can be actually represented as a normal form game because now we have the strategies sets for player 1 and player 2. But we will also show that that is not the most optimal way of representing this game because if you write down now the player 2 has a bunch of strategies and player 1 has three strategies. You can fill in the corresponding numbers in this matrix. Notice that whenever these two players are picking some strategy. So let's say player 1 is picking this strategy 0,2 and the second player is picking the strategy let's say RAA. Now this immediately gives the end of the game. It immediately says what is the outcome because the first player is picking 0,2 and the second player at the third. So it is essentially happening at this non-terminal stream. So 0,2 and then it is accepting it. So therefore the utility is going to be 0,2. So therefore if you look at 0,2 and the corresponding RAA you see the utility is being written as 0,2. So that is how this matrix has been populated. Now if you look at this game carefully you can come back and redo some of the predictable guarantees that we have given for this kind of games. And one of them were pure strategy Nash equilibrium. So we can find out a pure strategy Nash equilibrium for this normal form game. And there are few instances which I will point to so you can find out all possible Nash equilibrium. But you can see that there are certain outcomes like 2,0 and RAA is a Nash equilibrium, a pure strategy Nash equilibrium. Similarly this is also a pure strategy Nash equilibrium. So there are various such cases 2,0, RAA is also a Nash equilibrium. This is also quite interesting. So some of this Nash equilibrium that we have represented here are not really very credible. So for instance if you pick this example of 2,0 and RAA this is not quite reasonable because if it ever, so how can this RAA be understood? So it means that this player when it is at this stage is playing R and when it is at this stage displaying R and finally at this stage it is picking A which is the accept. Now why should a player, so let's say this sister will reject this offer if it ever ends up having ends up in this particular non-terminal state. It's always better because it is getting at least the outcome of 1,1 where it can get a new utility of 1 by rejecting it gets 0. So this is not really even though when you represent this in the normal form game you have this case where you end up having this particular outcome as a pure strategy Nash equilibrium. But that pure strategy Nash equilibrium is not really giving much information and in some sense this representation of the normal form game and its equilibrium concepts are not considering the fact that this is a sequential game and the sister or the second player has complete information about the current state of the game and it can pick the action accordingly. So that is what we are going to do so we are not going to use so for this kind of games where the information is completely available to all the players at every stage of the game. We will not transform them into normal form games rather we will go and find some other kind of solution concepts or other kind of equilibrium concepts that will be more appropriate for this set. Also I would like to remark that this representation of this extensive form game, this representation of transforming that extensive form game into a normal form game. It has a huge redundancy you can already see that the strategy space for all these players are actually blowing up and if there were more levels and more players then it would have blown up even easily. But EFG in that case is much more succinct representation of the game.