 So, what is the revelation principle? The revelation principle basically is that, so now we have, now let us look at, so this is the correlated equilibrium, now we can say let us, we can ask about how do we operationalize it? So, now give players a communication system, consider any communication system in which again players can send reports R and get messages M out of it. So, reports R i in R i send by the players and then they do which is, they get messages right. And now what would this lead to? So, if you equip them with this communication system once again I will write this out. So, if they have, so the problem for the players then is what reports should they be sending and what action they should be taking with the help of the messages that they get. So, the intermediate payoff that a player would get when he knows his own type and when all players are choosing you have a profile of reports R chosen and a profile of functions like this delta from messages. So, I am using the same delta because in a sense it is the same. So, delta I have used for deviation plan, it is also think of messages essentially as recommended actions that is why. So, R, delta this is what is chosen by players when they are two type, when the players two type STI what would be his, what would be the payoff that he would get from when there is a communication system like this available. So, he would, player I would choose actions, the players would choose actions based on messages received. So, you have delta 1 of m1 all the way till delta n of mn, true type is Ti. So, this is, so player has received, not Ti actually. So, so yeah instead of putting a given, given T and then you have this and then your, you have, so there is a summation over T minus I here and then the messages are being generated from the reports that are sent. So, the average has been taken by overall messages that can, that come out of the system. So, the whatever is the randomization that the communication system generates and over the types that of the other players. So, this gives you a payoff which is a function of the reported types and deviation plans. This is effectively now another game, a Bayesian game with a Bayesian game with types T. So, this is a Bayesian game with types T strategies, strategies gamma I which are just reports and deviation plans, not deviation plans, action plans whatever, what you take based on your message received. So, report RI and an action, everyone is clear? See this is, this is the, this is your Bayesian game with these strategies and parameterized now by the communication system. And you look for a Bayesian Nash equilibrium of this game now and it turns out that every such Bayesian Nash equilibrium can be simulated by a correlated equilibrium. The math is almost like self-evident. In fact, now once we have seen it, we have seen it earlier for the case where the, there was, you know, where players did not have types, the same thing exact same thing sort of works out here. So, we look for a, so look for a Bayesian Nash equilibrium Nash of this game corresponds to a correlation, correlated equilibrium. That means you look for a Bayesian Nash equilibrium of this game, you get, you can find a mu in it such that the same payoffs are achieved if mu satisfies the incentive compatibility constraint. So, this is basically your, this is what is called the revelation principle. In other words that all of this business of communicating etc., etc., is equivalent to players talking to a mediator, reporting their true and a mediator who implements a, a mediation plan in which it is in the players interest to reveal their true types. That is the revelation part. So, players communicating amongst themselves in a noisy fashion is the same thing can be achieved by players confidentially communicating with a mediator and the mediator implementing a true type, a mediation plan in which it is in players incentive to report their true types and obey the mediators recommendations. So, it is really amazing that such a thing in fact can be done, you know, to, if you think about it, how messy pre-play communication can actually get if you try to even model it like in a communication engineering type of way, but that there is such as elegant and neat theory and eventually the whole thing can be characterized by just some linear inequalities in mu. This is actually really an amazing, amazing thing. Now, there is, I should also tell you though that this is, you know, I mean as a general research comment. See, the reason we are able to do this so nicely and the reason, you know, this eventually all of this reduces down to a bunch of linear inequalities which are, you know, just a polydope and all that. The reason we are able to do this very nicely is because we are asking the right question. Okay. The question that, that is being asked here is what is the set of all payoffs that can be achieved? Okay. Under an arbitrary communications system, you know, you allow players, any, you equip players with whatever communication system, take the union of all the payoffs that can be achieved by them. That is what we are trying to solve for. We are not trying to solve for a more sharper question where here is a communication system. Tell me what players can do with it. Right. So, the union is much very easy to characterize, but any one particular point, if you I ask you that, you know, here is a, take this communication system, tell me all the equilibria that will come out of it. Right. That is not that easy to characterize. Okay. Because that is a messy, that is a messy Bayesian game. Basically, you have to, you have to solve, you know, essentially solve for all the Nash equilibria of that game in mixed strategies eventually. Okay. So, what the revelation principle actually does is it helps you characterize, you know, the puts a boundary on everything that can be achieved with a certain, that is in scope with the, with a certain set of resources. Right. But if you, if you give me a specific thing, specific resource, what, what can you do with it? That is a much harder question. Now, this is, this is often the case in research, you know, sometimes, you know, generalizing the problem can actually make it very easy, you know, rather than taking a specific instance. Okay. So, and, and a lot of game theoretic results have this flavor, you know, because you take a system level view rather than a micro level view of what is going on, you say, you try to say, okay, let us leave out the details and let us see what is the big picture that can actually be accomplished. Okay. And then that is the sort of view that this, these kind of results take. Okay.