 So, Amman's agreement theorem consists we took the situation where there were two players ok. So, we have an we had an Amman model of incomplete information with with beliefs and we considered an event A, suppose let A be an event and let suppose there were two players not denoted as capital this Roman 1 and Roman 2 ok. So, and the the statement of the theorem was as follows, so if the event that player 1 describes probability q 1 to event A is let A be an event and let omega be a state of the world. So, let if the event that player 1 describes probability q 1 to 2 event 2 event A is common knowledge in omega. So, it means the event that player 1 describes probability q 1 to event A, this event is common knowledge in omega and the event that player 2 describes probability q 2 to event A is common knowledge in omega then q 1 equals q 2 ok, so the this is what Amman's agreement theorem is about. So, Amman's agreement theorem assumes that we have a Amman model of of incomplete information with beliefs that means we are starting off with common priors, there is a specific event that we are discussing about for both players, players may have two different partitions. So, they may get information about about the chosen state of the world through two different media and through different levels of fidelity and so on. But there is agreement or there is the the the structures that lead them to to compute their beliefs about this event these structures are common knowledge. So, everybody knows that everybody knows that that that it is a dot dot dot that player 1 has computed probability q 1 for event A and likewise everybody knows that everybody knows that so dot dot dot that player 2 has computed probability q 2 for event A is this clear. So, what this means is all the the so everyone all everyone has there is complete agreement and complete sort of consensus on the states of the world in which such and such probabilities would get computed by these players. So, whatever are the underlying causes or the underlying states of the world in which the players would have ended up with these values those are are common knowledge. That event is common knowledge. So, the so the event that player 1 describes probability q 1 to event A is common knowledge an event that player 2 describes probability q 2 to event A is common knowledge. The question that Oman asks is the in this case can these two numbers q 1 and q 2 be distinct. Is it possible that players have started with a common prior we there is complete agreement on the circumstances in or the these the underlying causes in which players would compute a certain posterior probability for a and both are computing posterior probabilities for the same event right for that event A. In this case is it is still possible that the value of the posterior probability that they compute is different ok. So, what this means is players have started off with common beliefs they get different information ok, but the but there is complete agreement on the frameworks or the the methods or the processes they have for updating their information their beliefs can then then still end up with two different beliefs ok. Now this is this is this is actually this is actually an interesting question because players are getting different information ok, but there is agreement on the process they are getting different information it could be trivial if players were getting the same information obviously their posterior beliefs would coincide they are getting different information the partitions of both players are different they are getting different information each is observing the way the world in their own different way, but they have a way to agree on the methods that each will apply to compute their posteriors right. So, they have a way to sort of to to reconcile each others different points of view when computing the the the posterior distribution of this particular event ok. Then then is is it possible that they still have two different views about the event after getting the information. So, before getting the information they had common priors. So, which means they had the same view the probabilities that the prior probability of A was the same can the posterior probabilities be different if there is agreement on the on the process alright and Oman's agreement theorem basically says that it is not possible it is not possible that these two be different ok. So, in other words if you see two players disagreeing ok disagreeing does not mean that they have to fight say for example you know in a financial market the there is always a buyer and a seller right. So, the reason there is someone willing to buy and willing to sell at a particular price is because there is a disagreement on where the price will be the next moment the buyer feels the price will be higher or you can say the he has he has a a probability. So, let us say player 1 is the buyer he has a probability q 1 which says let us say q 1 is greater than half he has a he puts a higher probability that the on the event that the the stock will go higher that the price will be higher at the next instant. The seller has a has a higher probability that the stock will go lower these are two different players who are talking of the same event right, but computing two different probabilities for that event right. So, this so the the the question that Oman asks is well what is the cause you can say in some sense of of this disagreement why is there one fellow who thinks the price will go lower and one fellow who thinks the price will go higher and he according to so what this theorem basically narrows down to possibilities either you started off with different priors or you disagreed about how you are supposed to interpret the information you are getting what what do you mean by what is in enforce in this example so that is what I am saying. So, so if if these two probabilities are not equal then at least one of these has to break down either the common knowledge nature has to break down or the fact that they started with a common prior has to break down. So, the fact if they started with a common prior which means they did not agree to begin with or if they if they started with a common prior, but but yet are coming up with two different posterior probabilities which means that there is there they somehow do not see I to I on the process by which they are to recompute their probabilities or the scenarios in which they would each lead to a certain computation of the probability. So, so, so if you if you find for example, two analysts of two stock market analysts coming up with two different recommendations and one of these two is that is at play here either they interpret the events incorrectly or they just did not agree to begin with when in the sense that they were biased to begin with they interpret the events differently in the sense that they they both somehow at some level there is something that the other that we cannot say that this play this analyst knows that the other analyst knows that the other analyst knows and so on. So, there is no complete agreement on how we are how they are to arrive at the new posterior or what are the scenarios that could have led to that that are leading to this particular posterior. So, so, let us let us do the proof of this the proof is not that yeah, yes, yes, so process is based so process is based rule, but so that is just the formula part of it, but there is also the scenarios. So, what are the omegas in which base rule leads to this numerical value of the probability those set of such omegas that event is whether that is common knowledge is the question. So, so are they yeah essentially that is the so that that event being common knowledge see as I said events are common knowledge when they are publicly witnessed by players you know that is one of the most common way so, so there is so, so this event so becoming common knowledge would imply that they are both simultaneously or simultaneously and concurrently experiencing the update of or the obtainment of new information and then update of the probability alright. So, let us let us do the proof of this. So, if you remember so we will be using the structure of the of sets that are common knowledge. So, if you remember we had created this graph of the on the set on the states of the world in which there were there were edges if there is at least one player who cannot distinguish between two states of the world and we have we say we showed that there is a minimal set that is common knowledge in every in every state of the world and that set is the connected component of the graph that contains this state of the world. So, so let C of omega be the connected component containing omega it is a connected component of that graph and now remember also from the proof of that of this result we had also shown that C of omega is common knowledge in omega is common knowledge in omega that is one. Moreover, C of omega can be represented as a union of elements if I from this fancy FI from the partition of player I and this was true for each player I. So, for every player C of omega could be represented as a union of elements of his partition ok. So, let we can actually write it like this. So, that means that C of omega is equal to union over j Fij where j index is index for the element of elements of FI the partition ok. That means in other words Fij is in this fancy FI for all j. Now what is the we what we now want to do is we want to we are let us calculate the posterior probability of A. So, I will maybe draw a diagram here. So, this is y and this is one partition and this is let us say another part I am deliberately drawing it in this particular way and let us say the I am going to shade this with green this is my. So, this here. So, let us say for example this point here is this point here is omega the green region the green region here is C of omega and the way the reason I have drawn these partitions in this way. So, that you can appreciate essentially that C of omega is now the union of elements of partitions from both the red partition as well as the blue partition ok. So, now we what we know is that this fact is common knowledge that player 1 as scribes probability q 1 to event A. This is common knowledge in the fact that player 1 as scribes probability q 1 to event A this is common knowledge in omega ok. So, what does this mean? This means if you look at this this event omega such that the probability of A given f 1 of omega is equal to q 1 this is common knowledge in omega. Now, this being common knowledge in omega means that this event being common knowledge in omega let us call this let us call this event q 1 ok. e 1 is common knowledge in omega which means e 1 is e 1 contains C of omega is this clear? So, an event is common knowledge in omega if and only if it contains the connected component right ok. So, now what can we say about event A relative to C of omega? So, how is A look oriented relative to C of omega? Let us try to write the following first ok. So, first let us so what is this this p of A given f 1 of omega is equal to q 1 and now e 1 is this event e 1 is common knowledge in omega which means it contains C of omega. C of omega itself is comprised of the union of the union of several elements of partitions of player 1. Now, let us take any one partition let us take for example, a partition f 1 j ok. And you tell me how what is p of A what is p of A given f 1 j? So, remember f 1 j this this is what we we wrote right C of omega is the union of elements of player 1's partitions player 1's partition. Now, f 1 j is one of those one of those elements then what is p of A given f 1 j? Why is that equal to q 1? So, f 1 j is one of these green one of these elements which are covered by the green region right as it is that ok. So, since it is one of the elements covered by this green region what this means is that you know so say for example, this. So, this here for instance or let us take this this for the red guy say for example, this could this is f f of omega ok f of omega is this one. But you take any element outside this here in this green region the C of omega for that is going to for the connected component for that one is also going to be the same right. And so therefore, this event e 1 is going to be common knowledge even in that state of the world the other state of the world here ok which means which means what? Which means that even in this state of the world here player 1 will give probability q 1 to event event A ok. So, this is equal to q 1 this is equal to q 1 for all j such that right. So, for every element of player 1's partition which is contained in this connected component this is equal to q 1 ok. So, now let us write this out this is actually rather simple p of so this is equal to q 1 ok. So, now this is so I will take this on the left hand side now so far so far just an application of the formula. So, the point is this e 1 being common knowledge means that in all these states of the world player 1 will still compute probability q 1 ok. So, it is every state of the world in C of omega is going to still compute probability q 1 for event A ok. Now this so using that I have just expanded this out this way. So, all I need to now do is take summation over j where j ranges over such that f j f 1 j is a subset of C of omega this is equal to this ok. Now what is the left hand side? So, the left hand side is probability of A intersection C of omega how did I get that? The reason is because these elements of the partition are disjoint and their union is C of omega all right. So, therefore the left hand side becomes A probability of A intersection C of omega what about the right hand side exactly. So, it is probability of C of omega. So, this is q 1 times probability of C of omega. But then so which means what which means we have A intersection this is equal to q 1. Now what have we shown exactly? What did what have we ended up showing? We started off with this right and this was true for every j for every every point in this green region for every element in this green region I expanded that out and then summed it up over j and I got that q 1 is equal to p of A intersection C of omega divided by p of C of omega and what is this term? This left hand side term the main whatever be its value the point is its independent of the player. I started off writing this for player 1 and I got that q 1 is equal to this if I started off writing it this for player 2 I would get q 2 equal to this the same thing in other words q 1 has to be equal to q 2. So, therefore, we have concluded that this point here is this left hand side this is independent of the player which means that q 1 is equal to q 2. This is I mean in some ways you know in hindsight almost seems trivial the proof, but you know the point of major theorems usually is not the proof but the statement you know coming up with the statement is the hard part to come up with to guess what is it that you need to prove not what is how to prove it is the hard part. So, this is this is one more one example of that. All right, so this is Oman's agreement theorem. Any questions about this? Yeah this even being common knowledge. We have basically used that C of omega can be written as a union of partitions of regardless of what player it is a player 1 it is a union of his partitions player 2 it is a union of his partitions. So, no matter how the partitions are themselves arranged C of omega somehow ends up becoming a union of their partitions and that is basically what has been used. The common prior is P the this P here this is the or whatever this all of this is the prior probability that they begin with. Now, this being the what will fail exactly is this for all j. So, this here so this being not being common knowledge will make this so I mean there are approximate versions of this theorem also for example it is you can have you can relax this the common knowledge to something being some approximate versions of it of course this is this is fairly now fairly old it is about I think 80s if I am not on 80s or 90s. So, yeah there are various various you know sort of near versions of this where you have where you will get Q1 is almost equal to Q2 and so on.