 Welcome everyone. Let us recall what we were talking about in the last class. We had started talking about what is called the Oman's model of incomplete information. What is this comprise of? The Oman's model comprises of three different elements. The first element is the state of nature or the states of the world. This is what you can say is the true configuration of the world that we are modeling comprises of a set of players 1 to n. And for each player we have a partition of the states of the world. So, this fancy fi was a partition of capital y for player i. And the understanding was that a player can only know the state of the world up to the granularity that is described in the partition. So, for instance here in this case this omega here was the true state of the world. And the player who had the black partition for him the he can only know omega up to the up to the detail that is available in this element of the partition that he know what it what that means is he only knows that one of these yellow states is the true state of the world but does not know which one. Likewise for another player let us say player j he could potentially have a different partition in this case these are blue colored boundaries as are were his partitions and when omega is the true state of the world all he knows is that one out of this green shaded region is the true state of the world but not which one with specific one. This was the limitation that players had about knowing anything about the world. So, this is how they would this is the the the the coarseness to which they know what the actual state of the world is. Then from there we then said let us we took this example of two players where John and Paul we knew that the everyone it was John was colorblind and everyone knew that there were two possibilities for Paul that he could be colorblind or not colorblind but only Paul knew that whether he was colorblind or not. And there were three cars that they were looking at potentially which could have one there were three cars in the race and one of them could have won the race the colors of the cars were purple red and green and the definition of a colorblind person that the was that the colorblind person cannot distinguish between red and green. So, with this definition we decide we said let us let us try to understand what the partitions for the various players are. So, for John for instance it meant that because John did not know if Paul was colorblind or not John could not distinguish between these two states which is colorblind and purple and not colorblind and purple he only could tell if the car was purple or not purple. So, he could not distinguish between this these two states and likewise he could not distinguish between any of these four states which is the four the four states here capture whether Paul was colorblind or not and whether the red or green car won right. So, all four combinations are present here and he could not distinguish between any of these. So, his partition therefore of the states of the world which comprise of these six states were partitioned into two sets for John into this the first set here and the second set here. For Paul the states of the world were partitioned into 1, 2, 3, 4, 5 different subsets because Paul had a different way of knowing about the world. So, Paul could not distinguish when he was colorblind he could not have distinguished between the red whether the red car winning or the green car winning, but he could have told if the purple car was winning. So, he could have when the when he was colorblind and the purple car won he could he could actually say he could know that the purple car won when he is not colorblind he can of course distinguish between every possible color. So, that is that therefore gives you these three separate sets right. So, this is how we set up the partitions for this particular example. We then from there went to what it meant for a player to know about something first what is that something that we are that that a player needs to know. So, that something is what is what we called an event. So, an event is simply a subset of why when we are talking of knowing things we are talking of knowing about events. So, it event is then therefore a collection of the states of the world and you know an event if you know that any one of them has happened if you if you know for sure that that at least one of them has occurred. So, what does it what does that mean? So, here in this case we took this event A as this event what is this event stand for it was the event that Paul is colorblind and the green car as one or Paul is not colorblind and the green car as one or Paul is colorblind and the purple car as one right it is one of these three possibilities this for us was an event. So, which succinctly you can say that either the green car one or Paul is colorblind and the purple car one this is what this event captures and the and then we said what do these players know about this event in each in a particular state of the world. So, we fixed a particular state of the world which was omega star in this case the omega star is that Paul is colorblind and the red car is one and then we asked what does each player know about this event and our definition of knowing was that we said player I knows event A in state of the world omega star if you look at the partition that of the element of the partition that omega star lies in for player I that element is completely enclosed in the event A. So, you have these various partitions and every omega belongs to one element of this partition right. So, in particular for the yellow for the for when this is your true state of the world omega for the black partition it omega belongs to this yellow this yellow region here and for the blue partition it belongs to this green region here. So, what we know so what since a player knows that actually one of that element has occurred but not which one if that if that element if that entire element is a subset of A then we say that that player knows A ok. So, what he what he knows is therefore that he definitely knows that certainly one of these has occurred and therefore from there he knows that at least one of the elements in A has occurred ok. So, and therefore he knows A ok. So, so Fi of omega remember is the element of the partition that contains omega ok. So, remember this definition as well. So, this is where we ended last time. So, this gives us basically a framework for saying when does player I know an event in a certain state of the world ok. So, you have a certain state of the world omega star you have a player whose who has a certain partition which tells him what he could know and then from and you have a certain event to which we are referring. So, when does player I know that event in the state of the world omega star that is answered by looking at this inclusion. So, he knows player I knows event A in state of the world omega star if Fi of omega star is a subset of A. So, based on what we have discussed so far we can talk of questions like this we can discuss when does does player I know A in state of the world omega star right this kind of thing can we can be discussed. But we said that we would also want to discuss something more complex like for example, does player I know that player J knows something right these are the sort of things we would want to ask like for example, does Paul know that John knows that Paul knows the color of the car and so on. Now, how do we answer something like this for this that is where this framework is so elegant that it comes up with a with in one shot away for you to generate not just one level of knowing or knowledge about a particular event but also about what others know about that particular event and so on an entire hierarchy can be generated from this very easily. So, define what we call this is what is called a knowledge operator. So, let A be an event and let I be a player now we define Ki of A as follows. Ki of A is defined as the states of the world omega in which player I knows A. So, it is all possible states of the world in which it is true that player I knows A. So, this is set of states of the world player I knows A. So, as you change A you will of course, keep getting different sets Ki of A. Ki of A is the set of states of the world in which player I knows A. Now, what kind of a set is Ki of A? So, if A is any event what sort of a set is Ki of A? Yes, exactly. So, if A is an event then Ki of A is also an event it is a subset of the states of the world. So, an event is simply a subset of the states of the world. So, if A is an event then Ki of A is also an event. So, then once it is an event I can again talk of for any other player I can ask whether that player knows this event. So, for example, I can now ask does player J know Ki of A? So, what does this mean? For example, when would player J know Ki of A? He would know Ki of A if f J in state of the world omega star. So, he would know Ki of A if f J of omega star is a subset of Ki of A. So, if f J of omega star is a subset of Ki of A then it means that player J knows Ki of A and but what is Ki of A itself? But what is Ki of A itself? Ki of A itself is the event that player I knows A. So, if p J, player J knows Ki of A in omega star then he effectively knows that player then effectively we can say that player J knows that player I knows event A. So, this is effectively so which would mean that player J knows that player I knows A. But now I can go recurse on this further. I can talk so effectively. So, now I can talk of one more level. For example, I can write KJ of Ki of A. What is this? What is KJ of Ki of A? It is also an event and what event is it? It is the set of events. It is a set of states of the world in which player J knows that player I knows A. The set of one second. States of the world in which player J knows that player I knows A. What was the question? Where are you referring to? Is the element of the partition for player I? That element has to be completely enclosed in A. Yes. So, FJ of omega star is a subset of Ki of A is simply saying that player J knows Ki of A in state of the world omega star. So, what is it that you what was the other thing? Yes. So, you can also write that. So, let me write that actually here. So, KJ of Ki of A is the set of states of the world in which player J knows that player I knows A. So, if player J knows that player I knows A in state of world omega star. That is basically saying that omega star belongs to KJ of Ki of A. That is actually this is equivalent. So, is this clear? So, therefore, now what this knowledge operator is basically doing is now from here you can simply you can create any length of such hierarchies of knowledge. What does player I know about player J knows about player L about player M and so on and so forth. You can create any lengths of these hierarchies and answer and have a framework to discuss all of them because all of this knowledge of something has been reduced to just one which has been reduced to simply an event. So, let us do an example further. So, let us say for example, let us take the simple event. The event is that the purple car one. So, color blind comma purple and not color blind comma purple. So, this is the event that the purple car has one. So, now let us ask ourselves. So, go back to the partitions that we had written. So, what are the states of the world in which Paul knows that the purple car has one? What are the states of the world in which Paul knows that the purple car has one? Paul knows that the purple car has one in this event. Why? Because this is a subset of the event A that I am referring to. He also knows that the purple car has one in this event. Because the element of the partition that this event lies that this state of the world lies in itself is a singleton element and that is a subset of A. Likewise, this state of the world is in a singleton element which is also a subset of A. So, the states of the world in which Paul knows that the purple car has one is both CBP and NCBP. In both of these states Paul knows that the purple car has one. In summary, Paul knows that the purple car has one when the purple car has one. Is that clear? Because the states of the world in which Paul knows that the purple car has one is actually the A itself is the set A itself. What about John? When does John know that the purple car has one? Again, when this is when the true state of the world is this and when the true state of the world is this? Because these are both the element of the partition that they would end up defining is this and that is actually A itself. So, it is a subset of A. So, therefore, in both of these states of the world John knows that the purple car has one. So, once again John knows that the purple car has one when the purple car has one. So, Kp of A is equal to Kj of A is equal to A. Now, what is this? There is something really interesting that has happened here that the states of the world in which these in which both these players in which each of these players know this event A is equal to A itself. So, now what would happen if I ask one more level of the knowledge hierarchy? When would Paul know that John knows that the purple car has one? It would be A itself and when would Paul know that John knows that Paul knows that purple car has one? It would be A itself. So, in other words, if I can actually construct this I can have any such say Paul knows that John knows that Paul knows that John knows etc etc etc of A that would be equal to A. Now, what does this remind you of? This means that the event A which is the event that the purple car has one is common knowledge. So, this event A is in fact common knowledge. So, this is an example of how certain things can end up becoming common knowledge. For an event to end up becoming common knowledge, it would have to be there for that this sort of recursion eventually ends up with an invariant set. I will define what it means to be common knowledge in a moment, but this is let us just to give you an example of how this whole thing works out because we started off with common knowledge as our motivation. So, this is an example of that. So, actually maybe let us just do that itself now. Let us just define also welcome everyone. Let us recall what we were talking about in the last class. We had started talking about what is called the Oman's model of incomplete information. What does this comprise of? The Oman's model comprises of three different elements. The first element is the state of nature or the states of the world. This is what you can say is the true configuration of the world that we are modeling comprises of a set of players 1 to n and for each player we have a partition of the states of the world. So, this fancy fi was a partition of capital Y for player i and the understanding was that a player can only know the state of the world up to the granularity that is described in the partition. So, for instance here in this case this omega here was the true state of the world and the player who had the black partition for him he can only know omega up to the detail that is available in this element of the partition. What that means is he only knows that one of these yellow states is the true state of the world but does not know which one. Likewise, for another player let us say player j he could potentially have a different partition. In this case these are blue colored boundaries as were his partitions and when omega is the true state of the world all he knows is that one out of this green shaded region is the true state of the world but not which one with specific one. This was the limitation that players had about knowing anything about the world. So, this is how they would this is the coarseness to which they know what the actual state of the world is. Then from there we then said let us as we took this example of two players where John and Paul we knew that the everyone it was John was colorblind and everyone knew that there were two possibilities for Paul that he could be colorblind or not colorblind but only Paul knew that whether he was colorblind or not and there were three cars that they were looking at potentially which could have one there were three cars in the race and one of them could have won the race the colors of the cars were purple red and green and the definition of a colorblind person that the was that the color blind person cannot distinguish between red and green all right. So, with this definition we said let us try to understand what the partitions for the various players are. So, for John for instance it meant that because John did not know if Paul was colorblind or not John could not distinguish between these two states which is colorblind and purple and non not colorblind and purple he only could tell if the car was purple or not purple. So, he could not distinguish between this these two states and likewise he could not distinguish between any of these four states which is the four the four states here capture whether Paul was colorblind or not and whether the red or green car won right. So, all four combinations are present here and he could not distinguish between any of these. So, his partition therefore of the the states of the world which comprised of these six states were partitioned into two sets for John into this the first set here and the second set here for Paul the states of the world were partitioned into one, two, three, four, five, five different subsets because Paul could call Paul had a different way of knowing about the world. So, Paul could not when he was colorblind he could not have distinguished between the red whether the red car winning or the green car winning but he could have told if the purple car was winning. So, he could have when the when he was colorblind and the purple car won he could he could actually say he could know that the purple car won. When he is not colorblind he can of course distinguish between every every possible color. So, that is that therefore gives you these three separate sets. So, this is how we set up the partitions for this particular example. We then from there went to what it meant for a player to know about something. First what is that something that we are that that a player needs to know. So, that something is what is what we called an event. So, an event is simply a subset of why when we are talking of knowing things we are talking of knowing about events so it event is then therefore a collection of the states of the world and you know an event if you know that any one of them has has happened. If you if you know for sure that that at least one of them has occurred. So, what does it what does that mean? So, here in this case we took this event A as this event what is this event stand for? It was the event that Paul is colorblind and the green car as one or Paul is not colorblind and the green car as one or Paul is colorblind and the purple car as one. It is one of these three possibilities this for us was an event. So, which succinctly you can say that either the green car one or Paul is colorblind and the purple car one this is what this event captures and the and then we said what do these players know about this event in each in a particular state of the world. So, we fixed a particular state of the world which was omega star in this case the omega star is that Paul is colorblind and the red car is one and then we asked what does each player know about this event and our definition of knowing was that we said player i knows event A in state of the world omega star if if you look at the partition that of the element of the partition that omega star lies in for player i that element is completely enclosed in the event A. So, you have these various partitions and every omega belongs to one element of this partition right. So, in particular for the yellow for the for when when this is your true state of the world omega for the black partition it omega belongs to this yellow this yellow region here and for the blue partition it belongs to this green region here. So, what we know so, what since a player knows that one of that element has occurred but not which one if that if that element if that entire element is a subset of A then we say that that player knows A. So, what he what he knows is therefore that he definitely knows that certainly one of these has occurred and therefore, from there he knows that at least one of the elements in A has occurred. So, and therefore, he knows A. So, Fi of omega remember is the element of the partition that contains omega. So, remember this definition as well. So, this is where we ended last time. So, this gives us basically a framework for saying when does player i know an event in a certain state of the world. So, you have a certain state of the world omega star you have a player who is who has a certain partition which tells him what he could know and then from and you have a certain event to which we are referring. So, when does player i know that event in the state of the world omega star that is that is answered by looking at this inclusion. So, he knows player i knows event A in state of the world omega star if Fi of omega star is a subset of A. So, based on what we have discussed so far we can talk of questions like this we can discuss when does does player i know A in state of the world omega star. This kind of thing can we can be discussed, but we said that we would also want to discuss something more complex like for example, does player i know that player j knows something. These are the sort of things we would want to ask like for example, does Paul know that John knows that Paul knows the color of the car and so on. Now, how do we answer something like this. For this that is where this framework is so elegant that it comes up with in one shot a way for you to generate not just one level of knowing or knowledge about a particular event, but also about what others know about that particular event and so on. An entire hierarchy can be generated from this very easily. So, define what we call this is what is called a knowledge operator. So, let A be an event and let i be a player. Now, we define Ki of A as follows. Ki of A is defined as the states of the world omega in which player i knows A. So, it is all possible states of the world in which it is true that player i knows A. So, this is set of states of the world player i knows A. So, as you change A, you will of course keep getting different sets Ki of A. Ki of A is the set of states of the world in which player i knows A. Now, what kind of a set is Ki of A? So, if A is any event what sort of a set is Ki of A? Yes, exactly. So, if A is an event then Ki of A is also an event it is a subset of the states of the world. So, an event is simply a subset of the states of the world. So, if A is an event then Ki of A is also an event. So, then once it is an event I can again talk of for any other player I can ask whether that player knows this event. So, for example, I can now ask does player J know Ki of A? So, what does this mean? For example, when would player J know Ki of A? He would know Ki of A if fj in state of the world omega star. So, he would know Ki of A if fj of omega star is a subset of Ki of A. So, if fj of omega star is a subset of Ki of A then it means that player J knows Ki of A and but what is Ki of A itself? But what is Ki of A itself? Ki of A itself is the event that player I knows A. So, if pj player J knows Ki of A in omega star then he effectively knows that player then effectively we can say that player J knows that player I knows event A. So, this is effectively so which would mean that player J knows that player I knows A. But now I can go recurse on this further. I can talk so effectively. So, now I can talk of one more level. For example, I can write kj of Ki of A. What is this? What is kj of Ki of A? It is also an event and what event is it? It is the set of events. It is a set of states of the world in which player J knows that player I knows A. The set of one second. States of the world in which player J knows that player I knows A. What was the question? Where are you referring to? Here. Is the element of the partition for player I? That element has to be completely enclosed in A. So, fj of omega star is a subset of Ki of A is simply saying that player J knows Ki of A in state of the world omega star. So, what is it that you what was the other thing? Yes. So, you can also write that. So, let me write that actually here. So, kj of Ki of A is the set of states of the world in which player J knows that player I knows A. So, if player J knows that player I knows A in state of world omega star. That is basically saying that omega star belongs to kj of Ki of A. This is actually this is equivalent. So, is this clear? So, therefore, now what this knowledge operator is basically doing is now from here you can simply you can create any length of such hierarchies of knowledge. What does player I know about player J knows about player L about player M and so on and so forth. You can create any lengths of these hierarchies and answer and have a framework to discuss all of them because all of this knowledge of something has been reduced to just one which has been reduced to simply an event eventually. So, let us do an example further. So, let us say for example, let us take the simple event. The event is that the purple car one. So, color blind comma purple and not color blind comma purple. So, this is the event that the purple car has won. So, now let us ask ourselves. So, go back to the partitions that we had written. So, what are the states of the world in which Paul knows that the purple car has won? What are the states of the world in which Paul knows that the purple car has won? Paul knows that the purple car has won in this event. Why? Because this is a subset of the event A that I am referring to. He also knows that the purple car has won in this event because the element of the partition that this state of the world lies in itself is a singleton element and that is a subset of A. Likewise, this state of the world is in a singleton element which is also a subset of A. So, the states of the world in which Paul knows that the purple car has won is both CBP and NCBP. In both of these states Paul knows that the purple car has won. In summary, Paul knows that the purple car has won when the purple car has won. Is that clear? Because the states of the world in which Paul knows that the purple car has won is actually the A itself, is the set A itself. What about John? When does John know that the purple car has won? Again, when the true state of the world is this and when the true state of the world is this because these are both the element of the partition that they end up defining is this and that is actually A itself. So, it is a subset of A. So, therefore, in both of these states of the world John knows that the purple car has won. So, once again John knows that the purple car has won when the purple car has won. So, Kp of A is equal to Kj of A is equal to A. Now, what is this? There is something really interesting that has happened here that the states of the world in which these, in which both these players, in which each of these players know this event A is equal to A itself. So, now what would happen if I ask one more level of the knowledge hierarchy? When would Paul know that John knows that the purple car has won? It would be A itself and when would Paul know that John knows that Paul knows that purple car has won? It would be A itself. So, in other words, if I can actually construct this, I can have any such thing Paul knows that John knows that Paul knows that John knows etc. of A that would be equal to A. Now, what does this remind you of? This means that the event A which is the event that the purple car has won is common knowledge. So, this event A is in fact common knowledge. So, this is an example of how certain things can end up becoming common knowledge. For an event to end up becoming common knowledge, it would have to be therefore that this sort of recursion eventually ends up with an invariant set. I will define what it means to be common knowledge in a moment, but this is let us just to give you an example of how this whole thing works out because we started off with common knowledge as our motivation. So, this is an example of that. So, actually maybe let us just do that itself now. Let us just define also