 We have discussed the games with perfect information so far and the perfect information means that every player has perfect knowledge about the developments in the game until that round. So it can see what is the current state of the game, what are the actions that has been played by each of the other players and so on. However that has a very limited practical use as we have discussed in the previous modules. It may not represent certain kind of games like the card games where you cannot really observe the complete state of the game. You cannot really see what the action that has been picked by certain players and so on. So we need some richer representation than the perfect information game because it does not capture all sorts of games. And also if you talk about the representations, the simultaneous move games. For instance, the normal form, the games that we have represented using normal form like the neighboring kingdoms dilemma or similar such kind of games. We cannot really represent that using perfect information. So for instance, if you go to this very classic example of this neighboring kingdoms dilemma, the fact that one player when it picks its action does not know what the other player's action was is not being captured by the perfect information extensive form games. So for instance, if we draw the graph in the following way where first player is actually choosing at this point it has chosen but it has not disclosed it. So it has chosen let's say A. Now player 2 does not really know which whether player 1 has chosen A or W. But how can you represent that with the graph representation that we have done so far? We cannot. So in order to incorporate that kind of a situation into the extensive form game as well so that extensive form representation becomes as powerful as the normal form representation as well. What is done is this two specific history. So the history A or history W. These are two states of the game which are indistinguishable to this player 2. And to represent that indistinguishability in this modified version of this extensive form game, we just connect these two histories with a dotted line. And these two histories together form what is called as an information set. So we call this sets as information set which means that this for this particular player. So we call this an information set of player 2 and represent that using I. So I of player 2 and this is the first information. So there could be multiple such information sets in later parts of the game. So this particular information set is where this particular player cannot distinguish which state of it is in. So this is certainly a more general representation than PIEFG. In particular you can always represent a PIEFG using this representation which is known as the imperfect information extensive form game. And you can always force all this information sets to be singleton which will mean that that is a perfect information extensive form game. So therefore IIEFGs are generally more general in its definition than PIEFGs. However for a specific normal form game as in the case before the IIEFG representation might not be unique. So for instance the same game can be represented in a different way. So where player 2 moves first and now player 1 has an information set containing 2 non-terminal histories. So this will be I11 and the corresponding terminal histories. So one thing that you might have noticed that if you are in the information set because the player cannot distinguish whichever non-terminal history it is in. The action that it can pick the action set that is available to it has to be the same. So because it cannot distinguish therefore why how can it actually play different actions. So in all the nodes in a specific information set you will see that the action set are the same. And this is true for the previous example as well. So this is the informal definition of the imperfect information extensive form game. Let us make it more formal using the notation that we have developed so far. So the imperfect information extensive form game already starts with a PIEFG. So the first part is just like the PIEFG. But it also has this additional information which is the information set information. What is that? So for player I, this II is a collection of multiple sets. So II1, II2, IIKI. So there could be KI number of information sets for player I. So this is a partition of all the non-terminal nodes where player I is the player. So this is what it means that we are looking at all the histories, non-terminal histories where player I is going to be the player. With the property that the action sets in all these histories in the same information set if you are considering two different histories. So for instance if we are looking at this history H and this history is H prime. Then at both these histories the action set should be the same and of course the player set has to be the same. That's how we have defined it. Now as we have already said this IIJs are called the information sets for player I. And capital II is just the collection of all the information sets of I. So these individual things, individual sets are called the information sets. So there could be first information set, second information set and KI number of information sets for player I. So there are certain differences with the PI EFG. So since the actions of an information set are identical, X can now be, so this script X which was the action set can now be defined instead of individual histories you can define it over the information sets. So in the very degenerate case the information set could be just a history which is very much possible. But in general the information set now is the lowest level of abstraction of the history for a specific player. So we can define the X over a specific information set for all the histories which are living in that information set because this set is going to be identical. And therefore we can also define the strategies defined over information sets. This is also very natural. So how does it compare with the previous one? So just remember that earlier we had the same definition. So SI, the strategy set of a player I is the Cartesian product of all the non-terminal histories where player I is the player. Now we are just changing it to all the information sets of player I. So therefore it will just be Cartesian product of all the action sets over all the information sets of this player I. So with IIEFG so we see that now we have a much richer representation with perfect information extensive form game. We are not able to represent the normal form games but with this extension IIEFG now we can represent the NFGs. However there is a word of caution that it is not a very succinct representation it is not very concise. So there are representations so NFG representation is the most appropriate for the simultaneous move games as we have said before. And IIEFG or PIEFG is more appropriate for sequential move games. But nevertheless IIEFG is definitely a richer representation than both NFG and PIEFG.