 So, you have now a game, so this is a Bayesian game, there are n players, they have strategies X i, utilities U i and types T i. So, this, this, this and the, this is the prior probability with which the types occur. This is a prior of occurrence of types. So, in other words nature chooses, chooses T in T with probability, probability P of T and this we will assume is greater than 0. So, that we can freely divide and all that. Now, here it is okay because this is part of the problem definition. In mediation there is no such necessity that you have to give everything a positive probability. It may be that it is bad, you may actually want to stick to a certain, certain set of strategies or profiles only, okay. So, now what is mediated, so again we will look at mediated communication in this case. So, you take mediated communication in this case. So, now players have private information, okay. So, because the game is of incomplete information players are being born with various types, alright. So, so mediator now asks, mediator asks each player for its type, okay. This is the first step. Then based on the types that have been received, so suppose the mediator receives types T 1 to T n, these are the received types. Recommends action, recommends actions x, which is x 1 to x n with probability mu of x given T, right, okay. Now, so this mu which maps your T to delta of x, this is your mediation plan. What is the payoff that a player i would get when his true type is T i and the mediation plan mu has been chosen. What would be this payoff? This would be, can you help me write this out? So, an action profile x has been chosen with probability mu of x given T, okay. Now, player i will get a payoff u, ui of x given T from that when the true types profile is T, the type profile of all players is T. But he only knows his own type profile, right, which is T i. So, he only knows his type which is T i. So, here I am, what I need to do is take some this over all x, because that is the, there is an expectation over the actions that are chosen. And some also over the types of the other players, because you do not know what, you do not know the types of his or the other players, right. You know your own type and so you sum over the types of the other player, okay. So, this is the payoff a player would receive, okay. So, let us get this clear. This is the payoff that the player received when the mediation plan is mu, okay. His own type is T i, alright. And types of other players are T minus i, correct. But there is something more. See, why did I, so I have a T minus i here in this probability, why? Because that is the type of the other players, alright. That is the probability with which the types of the other players have been realized. There is a T minus i sitting here also in, as part of this T. This here, remember this, what you have here, this remember is the reported type. And this is the true type. True type occurs with probability P of D minus i given T i. And that will affect your utility. But the mediator is going to recommend an action based on reported types. Is this clear? So, this is the payoff that players would receive if they, if all players report their types truthfully. And the player knows his own type and the mediator chooses a mediation plan mu. Is this clear? If players had chosen some, had reported some other types, then that is what would appear here. It would not appear here because this is in fact their true type. But it would appear here because the mediator would choose actions based on reported types. Is this clear? Okay. So, this remember is the reported type. This here is the true type. So, we are actually assuming that these are equal. So, this is the payoff under plan mu when players report types truthfully. And player i knows his own type T i. So, he has come to know his own type T i, does not know the types of the others. And the types have been reported truthfully. And the mediation plan mu is under process. Is being used. Then this is the payoff that he can expect. So, now this is obviously now hinting at a very important issue. One is we had a question of obedience earlier which was there before this. But there is which we saw in the case of a correlated equilibrium. But now there is also another issue which is the issue of lying. Players may misreport their types. So, the mediation plan is suggesting, is recommending you an action X. So, players can and but the mediation plan itself is based on types that you have reported that players have reported. So, players can deviate from this plan. Players can deviate in two ways. One is misreport their types and two, they can disobey recommendations. So, let us once again we will follow the same sort of template of analysis. And this was pioneered by Myersen. It is really incredible how nicely this whole thing falls in place. So, what is happening here essentially what we need is that so to so think of the problem of designing a mediation plan. If you wanted to design a mediation plan you have to tell players what to what to do. But then what you need to tell them depends on what they tell you. It becomes this guy it becomes a very complicated it just seems impossibly complex that how could you tell someone what they do what to do when they are in fact they could in fact lie. Fundamental to all of this is this idea of what is called the revelation principle. Revelation principle essentially says that you can you can bring this all down to players revealing their types truthfully and then obeying whatever you say. Now there is actually revelation principle technically deals with something much much more complicated. You could you can in fact here I have asked players to I have said you can just reveal their types. Why do they ask them their types they can you can ask them something else also right. So, imagine for example if your type is let us say what talent you have some musical talent let us say. Now I could ask you something as a function of that. For example, I can ask you to sing something that that will that is a function of your musical talent or some or ask you to you know or something else which can be derived from that form that. So, this actually leads to the following. So, as part of our designing a mediation plan you have to you know if you think about it that way you have to not just worry about players lying you have to also worry about what exactly am I should supposed to ask them what is it that I should ask them and then based on which means what is the space of signals that come from the player to you right and then using that then I have to recommend an action. So, take auctions for example the the essential thing that player that each buyer the his type is how much he values that item. But should I directly ask him what how much he values it and that maybe he will lie based on that and then decide based on that you know who gets the item and and so on that is basically recommend recommendations based on that or maybe should I ask them something more indirect use that to then decide who should be what action should be recommended to the players. And it is not it is not at all obvious sometimes it may feel that ask doing something indirect can actually help you get you know help you implement things better. But so but the revelation principle basically says that I mean does not matter okay does not matter you can ask whatever you wanted to execute with with something with a with an indirect signal can also be executed with a direct signal okay. So, essentially this is the key idea is essentially a generalization of what we have seen already here in where there was just one way communication from the mediator to the okay. So, now there is going to be two way communication and the incentive compatibility type requirements that will be there will impose that it is in players interest to report that to to a obey and also to report their two types okay. The next thing we will see is once again this kind of mediation simulates any outcome of any communication system in which players report send in their types maybe truthfully maybe by lying and then that communication system then recommends action sends messages and from messages you get actions okay. So, so this kind of mediation once again simulates any pre-play communication with players having private information as well all right okay. So, we are just building towards the revelation principle. So, now essentially to in order to do that we have so players could deviate in these two ways as I said now each mu induces a game in which so each mu induces a game in which each player must choose a type to report and a deviation plan form report form recommended action. So, let us write gamma i then so this is so he reports a type si and he has a deviation plan delta i okay. So, si is now in is in ti all right and delta i is takes the reported action and produces another action okay. The payoff then form now so this we need to write out carefully okay I want to see if you can so the payoff is now for a for this for this game with these for the game with these set of strategies now okay. So, players will play s comma delta all right and they would have they have types t okay for a given s delta and t what would be the payoff so this nothing player i does not know his type as yet okay this is just for a for a profile of types for a profile of s reported types and for a profile of deltas what is the payoff so let us write this so ui which is his utility utility is going to come from what is going to come from the action that he actually chooses what is the action that he chooses delta of x where x is the action that comes from the mediation plan right so the mediation plan is going to suggest a profile of actions to the players that is this now what is this mediation plan what is this action suggested based on reported types which so I need to so reported types s so I have s here not t okay so you have delta 1 of x 1 till delta n of x n all right and then this is did I have what notation did I have given right here given given what s or t given t where this is now the two two type profile right so let us put a given here okay and the sum is over x and n okay so this is basically now a game of incomplete information parameterized by mu with this payoff okay and this being the strategies and these being the types okay so so this is so this is a Bayesian game with types ti and strategies gamma i all right okay so now let us let me ask you this so then if all other players are honest and obedient okay so they do not lie about their types and they obey the the recommendations that are been made by the mediator what is the payoff that player I would get if he reports a certain type and has a certain deviation plan okay we need to basically ask this question because this is the essential question of obedience or whatever right so so what is the payoff that player so let us call this ui star so this is this is here I am writing the payoff of player I from reported reported type si deviation plan delta i when all others are truthful and obedient okay so let us write this so what is this going to be a function of now it is going to be of it will depend on the mediation plan okay and so when he when we are talking of reporting type si all right okay so this is and let me put this more so since we are talking of reported type si this is when he comes to know his own type when he knows his own own type so si itself has to be reported based on his own type right so we need okay so this will be a function of now mu which is the mediation plan both are okay it will be a function of player i's player i's reported type si okay it will be a it will function of the deviation plan delta i and given this is given his own own type ti okay when he is once he has come to know his own type so when he comes to know his own type he has to pick a reported type okay all right so what is this let us write this so first let us write out the payoff what is player i doing so let okay let us write first recommended action okay so you recommended action x actually I need more space here so so let us write our recommended action x action has been recommended based on what and so action is recommended based on the types of reported types of all players so player i is reporting si t minus i si comma t minus i others are reporting t minus i okay all right now based on this what action is being chosen now ui delta i of x i okay which is so that has been recommended based on what is the action he takes based on what has been recommended to him okay what are the what are the other guys playing x minus i they have they are obedient right so they are they are playing what has been recommended to them okay given the true types of all players so this is given t all right so now I need to do some averaging so first is I have a sum over the recommended actions so you have some of x and x he also comes to know his own type right which is ti so what does he know about the types of the others pt minus i given ti right so he just he knows that the other types have been realized with this probability so I need to also do this this is fine so this is the payoff player i gets when he reports his type as si when his two type is ti and has a deviation plant delta i and all others obey and report to their tribes truthfully all right so now we have once again the the incentive compatibility constraints constraints again demands what should it demand in terms of our notation so on the right I need to have the what he gets when others obey but he does not all right so when that is just this u star i what do I have on the left do I have this when everyone obeys which was which was this one right so payoff underland view when players report their tribes truthfully and actually I forgot to write here and obey take the recommended actions okay okay so so here so what I have on the left then is ui of u given ti so what you have on the left here is this is when everyone obeys and everyone is truthful then all players obey and all are truthful and this here is when all but player i obey and are truthful okay so this inequality what this needs to hold for what for all something you know what is the for all yes for all types that he could report and for all delta i yes for all delta i which maps like this okay what else whatever be his truth type right for all ti and no mu has to be such that this was this for all n yeah okay and for all okay so this is this is what is needed okay so so and so what we have that so mu then is a mu is a correlated equilibrium of of the Bayesian game if it satisfies incentive compatibility constraints so if it satisfies these incentive compatibility constraints this one then mu is said to be a correlated equilibrium.