 We mentioned this definition of common knowledge in passing and I kind of defined it this notion of common knowledge in a kind of informal way because I use words like event, I use words like know, you know what does it mean for what I said that okay player I knows that player J knows that something and so on. All of this was being was done in a kind of you know heuristic way okay so now what we will do is we will make this formal okay and we will we will talk about what we will make in a very formal way what actually do we mean when we say that a player knows something okay alright. So the typical sort of situation that we want to talk about is that there are many different possibilities in which the actual world can exist but different players have different pieces of knowledge about it okay and now from that knowledge they want to talk about not just about what the world actually what the state of the world actually is but also what knowledge the others have about it okay. So we need that is what this framework is going to help you so this framework will help you answer questions like what does player A know about what player B knows and then there and then recurs on that like what does player A knows about what B knows about what C knows and so on okay so I will give you a kind a simple kind of example suppose there are there are these two players John and Paul these are these two players and John is colorblind okay and I will define what colorblind is so John is colorblind and everybody knows that John is colorblind okay about Paul we do not know actually if he is colorblind or not so there are two different possibilities he may be colorblind or we may not be colorblind but when he is colorblind he knows that he is colorblind for Paul there are two possibilities colorblind or not colorblind but when he is colorblind he does know that he is colorblind and when he is not also he knows that he is not okay so but only Paul knows that he is colorblind or not now why are we talking about colorblindness the reason is because these two players are looking at a say a car race okay there are cars of three different colors cars of three different colors purple red and green there are these three different colors color color colors suppose they are racing okay and they are all they are both watching that finish line and they can see some car pass now based on their own impairments they will be able to say whether or not a particular car has won okay a colorblind person colorblind basically means that he cannot he cannot distinguish between red and green so colorblind player cannot tell the difference between red and green okay so which he cannot distinguish between red and green which means that if a purple car does win okay then that fellow will be actually able to tell that it is the purple car that has won but if it is red or if it is either red or green that has won you will not be able to tell which of them okay now let us say for instance you know this is some kind of you know based on this they are going to decide which car to buy or which car to bet on or something like that okay so this is part of this is eventually linked to some decision making at the end but we are not even talking about that decision making here we are just discussing who knows what okay so try to answer a question like this so suppose it is that the red car has won then who knows what about this so if the so John is colorblind okay John cannot tell he just knows that purple has not won okay now Paul there are two cases so just like I told you that the red car has won I have to also tell you what is Paul state whether he is colorblind or not colorblind so now suppose Paul is not colorblind let us take it for simplicity Paul is not colorblind okay then what does Paul know red car has won okay what else does he know now that he knows that the red car has won and he knows also that John is colorblind he also therefore knows that John does not know which car has won okay but he does know that John knows that the purple car has not won is this clear right okay anything else what does what does John know no not necessarily see John does okay let me ask you a pointed question does John know that Paul knows the color of the car he just knows that Paul knows that it is not purple but he does not know that the ball actually knows the color of the car why because Paul's colorblindness is known only to him okay but not to John John's colorblindness is publicly known it is common knowledge right okay so just now I just talked about we just asked okay first we said what do each of them know about which car won then we asked okay what does each person know about what the other person knows about the car that has won now I can ask one more level so what does John know about Paul's knowledge of what John already knows John knows that yes so right so John knows so if the red car that has won John knows that Paul knows that yeah no but Paul see Paul knows what does Paul knows that the that John does not know the color Paul is not colorblind so Paul knows that John Paul knows that John does not know the color okay does John know that Paul knows that John does not know the color Why? Yeah, John is colorblind that everyone knows, but let us okay, that he knows that it is okay. Yeah, so this is, yeah, this is trivially true because John is colorblind and everybody knows. What about the reverse? Yes, yes. So I have to, that is why I cannot, it would be recursive if I said that it is common knowledge and then try to define common knowledge, right. So that is why I am motivating this sort of, let us say, heuristically, okay. But anyway, the point is that before we get into more of these loops, the point is that we, this is, we need a framework for doing, for this kind of hierarchical you know, knowledge structures that we ask what does each player know of something and that something itself could be what another player knows, is this clear? Okay, so the very beautiful and elegant framework for doing this and that framework is what is called, is due to Oman, Oman's model of incomplete information. So that, so the model actually comprises of the following, okay. So first, what we have is a, let us say, let us call this set S or maybe, yeah, let us call it S. This is the, this is a set of states of nature, states of nature or states of the world, okay. So the world is in one of these states, you have n players here. Now the, we are going to assume that players get knowledge about the state of the world through some kind of observation, okay. So this is the kind of framework that we use in control theory, physics, information theory and so on, where the, the or you in your knowledge that you have is through an observation. Now what does this mean? You, this observation basically tells you up to a certain level of granularity what actually is the state of the world. If you have perfect observation, it will actually tell you the exact state of the world but otherwise it will tell you up to a certain level of precision what the state of the world is, okay. Now the way to model this is to say well that for every player i you have this calligraphic fi, this here is a partition, is a partition of the set S. So the S i used for strategy is not so let me just change this to something else I will just call this y, okay. So fi is a partition of y, okay, for each player i. Now I will tell you what this partition means, okay. Now what is, what this says is this, suppose this is your set y, this is all possible states of the world, okay. Now what player i, a partition basically means a division of y into subsets such that each of these subsets they are all, they are mutually disjoint and their union is y, okay. So you divide y into these, into disjoint subsets whose union is the full set y. Now when an actual state of the world let us say omega where it gets realized what a player, this particular partition is for player i here, okay. This is player i's partition. Now what player i knows is that the state of the world is one out of this element of his partition. It is one of, it is in this yellow region but he does not know which exact point in that yellow region, okay. That is what his, so the partition basically captures this ignorance. It tells you that, it tells you that what is the granularity to which a player can, player knows about the state of the world. Now so if omega is the true state of the world which is this point, player i only knows this. Now there could be another player j who has a different partition, okay. Every player has a partition and these partitions could in general be different, right. Because their way of measuring the world or whatever measurement apparatus they have, the measurement information channel they have, whatever it is, that is different and so therefore they get a different partition. What does this mean? That means they, their amount of, their knowledge could be about the state of the world could be different. For example, for another player, this could be a partition. So I have now again chopped up y into a different set of subsets whose, which are mutually disjoint and whose union is y, okay. The blue one is for another, for another player, player j. So this, where this is for, for player j. Now when this gets realized, when, when, now when the, the, the, when omega gets realized, what does player j know? Player j only knows that it is one out of this green shaded region, but not which one out of that, is it clear? So this is our way of expressing the knowledge of the player. Now why does this, why is this, why does this make sense and where does this partition come from? See this partition is basically, so this is, as I said it is like a control theory or physics or information theory type model, because we are, we are thinking about what does a player know is basically that player knows about the world when he measures it. Now for example, if you want me to, if I, if I want to know if, if I have fever, I have a measuring device such as a thermometer, I, I check my temperature, okay. The temperature is a particular output of my, this thing. What I can conclude is I am in one of the states that leads to that temperature, right. It could be that I have fever, it could also be that I have maybe some, something else or it could be that I have just run a lot, therefore my temperature is high right now, could be whatever, right. So the actual state that I am in is in one of these vials. My measurement device only tells me the inverse image of my, this thing, it just tells me, so if I have a measurement device like a thermometer, say thermometer basically will map all my, all my body states y to a temperature, let us say some 0 to whatever, 0 to, right. So and when I get a reading T as my temperature, what I know is that my body state is actually just one of these H inverse of T's, right. So if I get a reading as, if my, if I get my reading is 100, I know that it is one of these H inverse of 100. If I get my reading as 99, I know it is H inverse of 99, etc, etc. If I, now you can see that as I vary T, each of these sets is going to be disjoint because they could not have possibly given me the same temperature, right. These are disjoint sets. As I vary T, actually what I end up generating is a partition of my space y because I keep, for every T, I get a different H inverse of T. This generates a partition of my state y. It basically and it, it is, it partitions my whole space into subsets of that capture my level of ignorance. So all states, so in particular let us say this one here, if I take this one here, this corresponds to all those y's, this corresponds to all y's, sorry all omegas such that H of omega is equal to say 100. These are all those omegas for which H of omega is equal to 99 and so on, right. But if I go to another, another person who maybe someone who has a different measurement device, maybe a more accurate thermometer or some, she does some another test altogether, he can get a bit different reading and a different, different partition, okay. Different reading means not just a different numerical value but also more accurate potentially reading or a reading that gives him more visibility into some states that just my temperature based measurements was not given, correct. So this is essentially our way of modeling knowledge in a game, okay. This is exactly how we model knowledge or information even in a control setting because there also you get what you get is you have a certain state, underlying state let us say x and what you measure is some function of that state, okay. This is a usual measurement equation where you get the output that you get is equal to some function of that state. So what you can tell is that this when you get a particular measurement or a particular output all you know is that the state is such that it would have led to that measurement but you do not know which exact state, right. So it basically partitions the state space based on the values of the measurement, okay, alright. So this is, this is, so this basically is our model for, this is our underlying model. Now from here how do we conclude what each player knows, we can talk about that but before that let us actually do, let us try and do an example. The previous John and Paul example let us try to model it in this way, okay. So now in that case you remember there are three different cars purple, red and green that could have one and three cars purple, red and green and there were, there are two possibilities as far as Paul's color blindness is concerned, right. So there are, that means Paul is color blind or not color blind. This is for Paul, okay. So now in the, in this case therefore the state of the world is in six possible states. The six possible combinations are this. So why then comprises of these these six possible combinations you have that Paul is color blind and the purple car is one, Paul is color blind and the red car is one, Paul is color blind the green car is one, Paul is is not color blind, purple car has one, all is not color blind, red car has one, all is not color blind and the green car has one. These are the possible states of the world. One of them is the going to be the true state in which we will be analyzing the game but these are the possible states right. If these are the possible states then what what does each player know? Let us try to let us try to understand that. So let us look at John's partition ok. Now what is John does not know ok if Paul is color blind or not and also John can John himself being color blind can only distinguish between whether the purple car has one or not one ok. Now what we will do is we what this means is that if you look at this this partition that you have for that I have drawn here it should have a structure of this partitioning should be such that all those states that John cannot distinguish between should be in one subset right. The states that he can that he definitely has a way of distinguishing will have to be in a different subset ok. So which now out of these states which ones can John not distinguish between. So let us we have to be so one of course at one level he can say that Paul is color blind or not sorry he cannot he does not know if Paul is color blind or not right. So then therefore you can say well all color blind non color blind can be put in one set but then that is not fine enough right because he knows a little bit more than that. So you have to go to the exact definition of the state see the state here says a combination of Paul's color blindness yes no and the car. So what combinations now can he distinguish between and what combinations can he not. So let us let us go let us let us suppose when the purple car has one he can tell that the purple car has one right but he cannot tell if Paul is color blind or not. Now what does that mean that means he cannot distinguish between this state which is color blind comma purple and not color blind comma purple right. So these two have to be in the in the same subset. So this must so there is a subset here which I can form by using these two states color blind comma purple and not color blind comma purple ok alright what else now all the remaining ones what do the all the remaining ones capture it is either red or green that has one and Paul is either color blind or not color blind it has no way of telling you know any of them apart right when green wins and Paul is color blind versus when red wins and Paul is not color blind all of them are for him they are alike ok. So all of these are in in one one this thing which is so color blind comma red color blind comma green not color blind comma red not color blind comma green. So you can see this is now his partition what what what do I mean by the partition here the entire space y has been divided into two sets this is one set this is another set they ok they are disjoint and their union is y ok. So what and and when one of these occurs Paul John can only say about which of which partition has actually occurred but not the exact state in that part for example if this if this state comes up he only knows that one from this box has come up but not which exact one ok ok. What is the corresponding one for Paul? So Paul can tell whether he is color blind or not ok what does this mean see if yeah so so if if since Paul knows he is color blind or not then you cannot have the following situation where you have a particular car has one the same car color but color blind and not color blind come and put in the same set right you cannot have a combination like this for example you cannot have a combination where color blind comma purple and color blind comma not purple in the same set because Paul knows whether he is color blind or not he should have been able to distinguish between these two sets ok but of course when he is not color blind he may not be able to distinguish between the colors of the car which is a different matter that is not what I am talking about right so you cannot have for example color blind and red and not color blind and red in the same set ok so what exactly is his partition let us let us write that out so let us let us go let us go one by one so color blind and purple when he when he is color blind and the purple car has one can he tell exactly that ok so he knows so this set is a singleton set by itself so when this happens he knows exactly that that has happened ok what about color blind comma red so he cannot distinguish between color blind comma red and and color blind comma green so when he is color blind he will not be able to distinguish between these two ok so these and color blind comma green are in one set when he is not color blind he can distinguish all of them all the all the car colors so all of these are then in separate sets right so this is now Paul's partition ok so when when any of these occur he of course knows what the state of the car is likewise this one also when this one occurs he only knows that one of these is ok alright so I will just set up some more notation then now that we now that we know this so the way we will analyze further is that what we will we will pick a particular state of the world ok and that state of the world is known to us as observers ok but the players may not know ok so this is what so a situation a situation so what I wrote out above is the model but a situation of incomplete information comprises of a particular state omega star n y ok now when omega star occurs what do players know so the let let us define this so this simple f of omega this is the element of this fancy fi that contains omega so this simple fi is the element of this fancy fi that contains omega ok from that so is the element of the partition that contains the actual state the particular state ok so when so in a situation where omega star is the actual state of the word fi of omega star is what players know that is the level of knowledge ok ok now the players will be talking about we were rather we will be talking about knowledge of players of events ok now what is an event an event is is simply a subset ok so you take any subset of y that is that is what we call an event ok and now we can discuss of what two players know about this event and so on ok so for example let us take this this subset the subset is that let us say color blind comma green not color blind comma green ok so can you tell me what is this what is this event example in words can you tell me what this what is this event referring to see an event is a is just a collection of states of the world and the event is when you say when you have this collection it is an or between those those those states ok it is either this or this or this that is what we are calling as even so so this here for example this event a here is the event that either Paul is color blind and the green car one or Paul is not color blind and the green car one or Paul is color blind and the purple car one ok so in short I mean I can summarize this is it is the event that either the green car has one or Paul is color blind and the purple car one ok that is that is this event ok so this is either green car one or Paul is color blind and the purple car one so be careful and if I am putting this in brackets because I want to make sure the and is between those two last two last two things ok alright so we now this is an event and we we are talking of an event in us when when in a context of a certain situation right there is a situation means a particular state of the world has been chosen and now in that context we want to talk of a particular event now it is possible that this event itself is not occurred in the particular state of the world right for example if they if the if the if the situation that we are talking about is that Paul is color blind and the red car has one if that is omega star so omega star is so if omega star is is this sorry color blind and the red car has one then this event has not even occurred right because none of the possible thing states of the world at that event refers to is is is equal to omega star ok so the two state of the world is is omega star but the event that you are referring to is not has not occurred in this state of the world ok so so that is why we need another terminology we say that an event that an event occurred actually occurred is a bad word in English because it seems like it is it you know it is also known that it has occurred so that is why the word used is event we use the word obtains we say that an event event a obtains in omega star if omega star belongs to it ok so you have a state of the world and we say that an event a obtains in the state of the world if that state of the world is present in is is is an element of that event ok ok so obviously it makes sense to when you are in a particular situation that means you are discussing a particular state of the world as the two state of the world it makes sense to discuss about events that have actually occurred in that right so the others that are not occurred or that have not obtained are kind of are you know kind of hypothetical right in some sense that they are there they could have occurred but but they are not what is being considered in this situation ok so I will end with one last definition so we say player i knows event a in state of the world omega star f i of omega star is a subset of it ok so we say that player i knows that that event ok or whatever knows that event has occurred or knows that whatever that event represents in a state of the world in the situation omega star if f i of omega star is a subset of it what does this mean it means that go back to what f i of omega was it was the element of his partition that contains omega star right is the element of his partition that contains omega star now if that element so that so all his ignorance about omega star is captured in f i of omega star so he knows this that the state of the world is one of f i of omega star but not which one now if that is a subset of a ok then it essentially tells him it tells you that what it means is that he knows definitely that a has he definitely knows a ok because he knows form from his information he knows that he can conclude that one of a has occurred for sure because he knows that one of f i of omega has occurred and f i of omega star has occurred and f i of omega star is a subset of it this so that is our model for knowing a particular event all right