 So, now let us make this formal, so in short, in a game of mediated communication, the player communicates separately and confidentially with the mediator. After hearing the echo, the recommended action is free to do, to choose any strategy. So, now what is the payoff that a player can get if he follows this recommendation? So, suppose mu is the correlated strategy, what is the payoff that player i can get under mu? It is ui of mu which was, which is simply this, this is payoff under mu. Suppose if he instead decides that after hearing his recommendation, if he decides to deviate to something else, then what would the, how would we model that? Well, then what he would get is, so he, suppose player i sees, player i gets recommendation xi and from there, so then he disobeys, so he disobeys, so then he will change that, map that recommended action to a gamma i of xi, so this is the deviation, so what you want is that the payoff form following the recommendation has to be better than that from any deviation from that recommended, assuming that others continue to follow it, so what is this, so the payoff, so what we want is ui of mu should be greater than equal to mu of x. The payoff that player i will get from deviating to, to many lessons, many here is xi, he deviates to gamma i of xi, while others continue to stick to x minus i, whatever has been recommended to them and then outside you have an expectation with respect to mu. This has to be true for all gamma i and for all i and n, now you will see this reminds you of something, so what has happened here, player i has received a recommendation xi, from there all he knows is that others have been recommended posterior distribution, the posterior distribution of x minus i given xi, so essentially this is creating a game of incomplete information between the players, where essentially the recommended actions are there sort of like their types. So now here, so I can, because any deviation like this is allowed, I can actually write this whole thing in another way, essentially I can write what would I get if I followed the recommendation and what would I get if I just played something else, I do not need a function of that recommendation and all that put in there. So you can actually show that this is equivalent to, so ui of mu is greater than equal to this ui of, so now you can and this has to be true for all xi dash in xi and for all players i. Now you can see this portion here because there is a sum over all x and you have a mu of x here, what is there in the numerator effectively is the marginal of x minus i, and there is also a marginal of, there is a marginal, you can effectively then one way of writing this whole thing would be that you have a like a posterior distribution. All right, so this now is a form of, these are a form of again what has been a form of obedience constraints, what you know we talked of obedience in the case of the principal agent problem where the you know principal wanted the agent to do something but the agent may not actually be may not actually be following that this thing that was a question of moral hazard and so on. And there was this question of obedience essentially that came up, so here this is essentially a constraint that is imposing obedience. So a strategy through mediation can work only if it actually has satisfies this obedience constraint or another name for that is incentive compatibility or obedience. So any mu that satisfies the above is called a correlated equilibrium. So as I said you can now that this correlated equilibrium is defined, you can essentially it is the set of set of, it defines for you the set of payoffs that can be achieved through mediation. And suppose you are a mediator, you may want to you know solve a certain some other global objective for example, subject to that you want to achieve over these agents, like you have some set of agents that need to be incentivized, told what to do, but then what they do needs to what you tell them is something that is what they need to actually follow. So then under a correlated equilibrium this is what you can get. Now if you look at the constraints that define the correlated equilibrium, these are actually linear in mu. And this is larger than the set of Nash Equilibria or Landers convex hull of the set of Nash Equilibria. That is the beauty of this, that you can get more than the set of Nash Equilibria through this. So in fact I can now for this figure I can tell you what this looks like. So you have your points 1,5 here, just writing out the payoffs for the two players, 1,5, 5,1 and there was one point which was 0,0 and there was a point which was 4,4. So let us take this orange here, take this region, what is this region here, what is the orange region? Delta of X are the probability distributions, mu of mu where mu varies over delta of X. So these are the payoffs that can be achieved under arbitrary correlated strategies, not correlated equilibrium. Any correlations you create, this is the set of payoffs under those. Now this here, so this here, these are, this will be V2, you can show that. So remember we had those security levels for the players that players would want to, this is the minimum they need to be guaranteed in any contract in order to sign a contract, otherwise they would just play up your strategy and ignore all the contract, that was these levels. So the subset that satisfies those participation constraints is this set, is this subset then. Now let me see if I can put another color on top of this, that is your participation. Now inside this you have also the set of all, looks something like this. So you have inside this the set of all payoffs that are achievable under a correlated equilibrium. So this is what can be achieved under a correlated equilibrium. You have to look at these linear constraints. So the point of this is exactly that, that you take all these linear constraints, this actually defines a polytope, they are just linear inequality constraints. It is a polytope in mu, you then evaluate the payoffs that is a polytope in the payoff space. So that gives you this set. So this is the payoffs under correlated equilibria. This is payoff under arbitrary correlated strategy and then the green region here, this here is under participation constraints or payoffs under finding contracts. So this is how this looks. So here is the interesting thing now, now why did we bring in mediation at all? Because as I said mediator is just as I was saying is not there as a player and then once you if you want to think of it as a player then you have to think of trusting the mediator, not trusting the mediator etc. So why the reason we brought in the mediator is because mediator is just a sort of a thought device. What you can achieve with a mediator is actually equal to whatever you can achieve once you allow also for indirect communication. Now I will explain what indirect communication would mean. So indirect communication is essentially, so the earlier direct communication is where you can observe the sample and that is you can think of that is basically noiseless communication. You know exactly what has come out of the experiment. But if there is some amount of noise on top of that or some partial observation or some additional randomness or whatever or some incomplete something where you cannot completely observe what has happened then that amounts to indirect communication. So how do we model this? Now hopefully I will be able to finish this today. I want to finish this today. So how do we model this? So what we we can allow players to communicate via a communication system. Now again this is going to be implicit. Communication system means I am not going to write a multiplexer and all of that as part of the communication system. This is all implicit. So how does this work? Players provide, players give reports or rather not give send reports. These are essentially messages. Let us call them reports. Reports RI, I in N and let, so these reports lie in R i and let R be this product of R s. So players send these reports. There is a communication system in place which will generate messages that players will receive from these reports. So let us call this system nu. This nu generates messages m1 to mn. There are n players so m1 to mn given the reports that these players have sent. So it is just a conditional distribution. So this conditional distribution is your communication system, conditional distribution. This is your communication system. This is the messages received by the players. So players send reports R1 to Rn. It generates messages m1 to mn. Now in the case of direct communication or one-to-one communication, you can have a very specific nu here where nu will, you know what player 1 sends a report R1 and it is received noiselessly by player 2, suppose. So then nu will take the form that m1 is equal to R, m2 is equal to R1 with probability 1 and everything else is probability 0. That is the, you can model a kernel of the conditional distribution to model any such communication graph between the players. Now, so there could, this also allows for noise in the medium. And noise in the medium is important because we also want to allow for, you know, the kind of thing that the mediator was able to simulate. The mediator was able to give partial information or keep a little bit of vagueness about what was recommended to the other player that comes through in this model through the role of noise in the communication medium. Now, so what happens? So given such a communication system, what are the pure strategies of the players? Pure strategies of players. So let us call this gamma i. These are the pure strategies of the players. So the player has to send a report R i and based on the message that he has received, he has to take an action. So he has a function gamma i such that R i belongs to capital R i and gamma i has to just map whatever message he receives to an action that he needs, he takes. Is this clear? Okay. Now, what is the payoff then? Then payoff then becomes under this. Now, suppose players play R, gamma, then player i is payoff then becomes something like this. Is this fine? So this is the expected payoff that the players would get through by playing a pure strategy, by playing a profile of pure strategies R, gamma. When this is there, this here is your communication system. So what has happened as a result of this? Now, you can think of this in the following. This is basically created a game of a game in which these are now their utilities. So you get a game for a game where these are the utilities and you get one such game for each new, for each communication system. So each communication system generates a game in the R gamma space. The communication system new generates a game in the R gamma space parameterized by new. So now what we need to do is, so given a new, you can now look for equilibria in this. So this results in a game in the R gamma, the R, R and gamma look the same, R gamma space parameterized by new. So the pure strategies in this are R, gamma for the R I gamma I for the players. Now players can now choose an R, gamma at random and that would give you a mixed strategy in this game. So a report at random and a mapping that maps the messages to an action at random. So then let sigma be sigma I, a mixed strategy which is basically a probability distribution now on the pure strategies gamma I, on the R, gamma space. And now you look at the payoffs that would come under this by basically just taking the expectation with respect to sigma. So you have then you can write a payoff now in sigma, sigma minus I by doing this sigma 1 of R 1, gamma 1. So here is the important thing. Now you take any equilibrium of, so this is basically the main result in this. Take any equilibrium from that can come out of this. It is actually is equivalent to a correlated equilibrium with a certain correlated strategy. So what you can do is you take any equilibrium from here and from here form, so any equilibrium in these in these sigmas, a mixed strategy equilibrium in this game which has a, which is equipped with a communication system. Take that equilibrium from there you can generate a correlated strategy such that it is actually a correlated equilibrium. So in other words, a communication system like this generates basically a correlated equilibrium. So or equivalently correlated equilibria are essentially simulations of some way or essentially simulations of games like this which are equipped with communication system. So the set of all equilibria or the set of all payoffs that can be achieved by equipping players with arbitrary communication systems is equal to the set of all equilibria that they can get with under correlated strategies. It means the set of all correlated equilibrium. So how do I create a correlated equilibrium from this? You can just create a correlated equilibrium like this. You want to define mu of x in this way. You let players choose r, gamma in this. Essentially you look for the action that they would have. So all you have to do here is this is this may look a little intimidating but I will just write this for completeness sake. So remember what has happened here? You take a report, report reads to a message, message leads to an action. Now you are taking an average over the probabilities with which the reports were generated, the probabilities with which the messages were being chosen as a function and the message function was being chosen and then from there you are generating therefore a probability with which a certain action is being chosen by the players. That probability therefore is just this. So you are effectively getting a probability with which various players are playing various strategies. And from here you can basically you can just argue that this is in fact a correlated equilibrium and the argument is something again we have used before. If this was not a correlated equilibrium that means if so let us say there is a suppose of this equilibrium specifies a recommendation to some player and if it is not in his interest to follow that recommendation then it would mean that he would not follow his own equilibrium strategy sigma because he would have an incentive to deviate from what was his own equilibrium strategy. So if sigma is an equilibrium here then it generates a correlated equilibrium and the converse can also be shown in the same. So just summarize this. So if any equilibrium sigma of the game with communication system mu generates a correlated equilibrium. So here is something interesting actually maybe I will just mention this point. So what was the chain? You have reports, reports leading to messages, messages leading to actions and the reports are also being generated randomly as part of the equilibrium here. Sigma is choosing a r comma gamma randomly. So therefore what is really happening only thing that is left when you when I take an expectation like this out here the only randomness left is in the choice of the action. So it is the action which is random now. Report has been averaged over all the messages have been averaged over and you have also averaged over the choice of the pure strategy itself in the mixed strategy. So only the action is random and that is why this is therefore a distribution on the set of actions. Now here is the interesting thing that the correlated equilibrium is simulating a communication system in which players are communicating with each other but the correlated equilibrium itself is only recommending actions. So you see one something very interesting has happened here. Players in a communication system players were allowed communication between themselves whereas in a correlated equilibrium the communication is from the mediator to the players. So there is a one way communication between mediator to the players whereas here in the with the communication system there was effectively a two way communication because players are allowed to communicate with each other. So where did one of the directions of communication is somehow gotten lost here? Somewhere we have lost one direction of information exchange because now just the mediator is just recommending whereas earlier players were exchanging information with each other. So what is the reason that has happened? I know it is a little bit fast but I will tell you I will tell you an intuitive reason and it is really beautiful why this happens. The reason this happens is players have no private information. There is nothing private that they know. So there is nothing for them to tell the mediator. So this business of reports and all of that is just a way of creating randomness. It is not really exchanging information. So therefore it is enough that there is players do not have any types of their own. There is not a game of incomplete information. There is nothing for players to share with the mediator. So only one way communication from the mediator to this thing is enough. So only there is the only benefit that is to be derived is that of randomization or noise not of exchange of information. So as a result of that actually this thing is correlated equilibria actually end up simulating the equilibria of such games even if you allow for communication.