 So, we have discussed problems where players are allowed to enter into binding contracts and you had a set of payoffs that could be achieved under binding contracts. Now, suppose players are allowed to communicate, but there are there is no scope for getting into binding contracts. So, when they say that they are going to play a particular strategy, there is nothing to come to bind them to that right. In this case, what now you say you have a set of payoffs that can be achieved therefore, with communication, but without binding contracts. And then there is a set of payoffs that can be achieved with communication and binding contracts. So, which is the larger set, yeah, so the set of payoffs that can be achieved with binding contracts is the larger set. And the reason for that is because if a payoff can be achieved without binding contracts then the same payoff can also be achieved if the if that contract was binding right. So, so, so you have this inclusion that payoffs that can be achieved achievable with communication and binding contracts. This is a superset of payoffs achievable with communication only and this here is a superset of payoffs that are achievable achievable with only direct communication. And this in particular further is a superset of payoffs that are achievable under Nash equilibrium. Now this was also a point of confusion last time some of you asked me later about you know what exactly is the discussion distinction between payoffs achievable with communication and direct versus direct communication. Now I will, so today what I will do this a little more elaborately because I think the law it did not come out very well last time. Okay, so firstly remember all of these things right amount to pre-play communication. All right, so players are communicate so all the communication that is being allowed here is pre-play. So, before they commit to before they actually choose their strategy they are announcing what they want to do they are trying to you know influence the they are creating lot trees and or random draws and so on all of this is pre-play. But when it comes to finally going and choosing the action that at that time they did not then choose their actions independently all right. So even when we were looking at binding contracts when binding contracts were allowed right even here the problem where the players was to independently choose which contract to sign right the way you had a game initially with some set of pure strategies and it was augmented with the with additional strategies where the strategy was whether to whether or not to sign a certain contract right the choice of signing the contract and you could have any number of such strategies you could keep adding more and more contracts to the whole picture right. So, that choice of which contract of whether to play your original pure strategies or to sign a particular contract that was again being made independent all right. But then the contracts being binding meant that when you choose that strategy when you choose to sign it you are bound by that those by whatever other terms of that contract all right. Now payoffs achievable with communication alone again has the same flavor this is pre-play communication players are allowed to communicate pre-play but once the communication is done they go and choose their actions independently all right. So now what this means is so if you think about this what is happening is that so we had two frameworks here one was correlated strategies and the other was strategies with that we said we could augment the problem with the communication system that allowed players to communicate. Now in either of you know you can look at it in either way essentially what is in if you have a communication system then what is happening is pre-play players are sending reports the communication system is generating messages and then using those messages players are now going and picking their strategies all right. So all this pre-play communication is make correlating there correlating there you know their the strategic choices right. So now what effectively happens is so once the you can think of it you can think of it this way that the other framework was the one where there was a mediator and the mediator was had was recommending an action to all of these players and again once the recommendation is made recommendation is made confidentially to the to the players it is the players choice whether to follow that recommendation or not right. So therefore we impose that a for a for a for the for such a correlation to be to actually be a for a payoff to be achievable under such a correlation it has to satisfy the obedience constraints right. So that means once the once that is received once this the the recommendation is received it is in the interest of the player to follow the recommendation. So with recommendations this is kind of easy to understand now what happens in the case of direct communication. So here is here there is slight possibility of confusion now you can so what did we see here we said that under direct communication what players can achieve is the convex hull of the playoffs from the various Nash Equilibrium because they can essentially toss coins between toss a coin and choose one of the Nash Equilibrium based on the outcome of the coin. Now here also the this this this agreement to do this to do this randomization is you can think of it this way the I so I justified it in the following way last time where they said well whether to follow this agreement or to play some other strategy was the choice of the player right. You can also but remember here this is being conducted as a public lottery means that players get to view the outcome of the lottery itself right. So they get to know what what has come out of the coin toss and therefore they know what has been recommended to the other players right. So they know the exact thing that has been so now because this contract was not binding so what this means is that after the players can deviate from the from this even after observing the coin toss. So suppose it comes out heads and it turns out that a player has to play a particular the players are now according to this plan the for heads you are supposed to play a certain strategy what is this strategy would be something that players would stick to only if it was a Nash Equilibrium right. So after the coin toss so you think of the coin toss as again a kind of recommendation that has been made the outcome of the coin toss as a recommendation that has been made to the players. Players come to know that this is what we are supposed to play and there is no incentive to unilaterally deviate at that stage if that thing was a Nash Equilibrium right. So the point is the this strategy was created through communication through pre-play communication during play what happens is a particular Nash Equilibrium gets realized as the outcome of the coin toss and now question is whether players want to follow the recommendation of that Nash Equilibrium or not and there is no incentive to unilaterally deviate at that stage is this clear. So this basically puts it in the same framework as a correlated Equilibrium essentially the coin toss outcome is like your recommendation to the player yes. So the recommendation is the probability with which various things are various strategies are going to be recommended that is common knowledge but here the important thing is the outcome also is known whereas in correlated strategies players do not need to know the outcome they just only need to know the strategy that has been recommended to them all right. So you can think of this as the direct communication as a public lottery as a lottery whose draw is public we know the fundamentals that are leading to the various outcomes whereas a general correlated strategy is one where the lottery is not public the lottery is done behind the scenes you are just told a result that you should be doing this and you are told that confidentially right each player is told that confidentially ok is this clear and then finally we have of course Nash Equilibrium where none of these mechanisms are possible ok so that is the hierarchy. So in each of these cases you can think of it this way we essentially what this does is you take the original game augment it with a with some pre-play mechanism like this and then create a new game in which the study now in which players have to now act based on the information that is coming from the pre-play mechanism is this clear yeah yes yes yes yes yes now the final thing we which we discussed last time and which came out which was where we were talking about I was talking about this case where correlated equilibria I said simulate the outcome of any pre-play communication system right so we and so how did we model a pre-play communication system we said it is this it is some probability distribution like this where R is the reports this is a profile of reports that players submit to the communication system M is the set of is the profile of messages that the communication system generates so R I the report sent by player I and M I is the message received by player I and what he had to do was he had to pick a strategy so I use the notation gamma and I realized from once I saw it it looked like my R so I am going to use delta this time okay so so delta I which mapped the messages this is the set of messages to an action right now this is your communication system okay this is your communication system essentially it is just an input output model of the communication system R goes in M comes out you can model anything any more specific system using this so for example if your players are communicating directly on a bilateral way between each other that can be modeled as a specific case of this by taking you know so suppose player I is talking to player II then you can have that this communication system gives probability I only to those only to when player I is talking to player II right those kind for those sort of events and puts all mass on those sort of events and so on so this is fairly general this rather this is completely general it can it models all sorts of bilateral communication and more general communication also where reports come in and then some kind of device generates the messages okay so the main thing is in fact the randomness that is present in this okay this conditional random this random that messages gets generated randomly given the given the reports is actually what let us let you maintain that gives you the possibility of partial information to the two various players right because a correlated strategy we saw relies on partial information it may you need to tell a player only what he needs to know and is there is there is a vagueness about what the others have been told that noise or that partial information comes about because this medium is noisy right if this medium was perfect with if this communication system was perfect and no is less so for instance when when a certain message was received by player II he also knew what everyone else is being recommended or what everyone else is getting then then it would effectively start collapsing down to a case of direct communication okay so the so remember that the noise here is is the is is essentially what is allowing for correlation between the strategies of the place okay or rather mechanisms that are like correlated equilibrium okay now what I showed last time was that you a correlated equilibrium simulates the outcome of any such any such any such problem so what so what is the the form of the problem you now have the original pure strategies of the players which are these which are your excise players send these reports based on the reports messages get realized and the new strategies now for the players are what report should should they send and what should they do once they receive the message so this now becomes the new game and as I said after the pre-play communication the choices are now being done independently so players can now choose their report and their the this function this function delta I independently after that is this clear and we what we saw was that you take any equilibrium of this it correspond any equilibrium that comes from this game right where players have to now choose our eyes and delta is it is now a simultaneous move game in this space alright what the any equilibrium of this game is and a correlated equilibrium as defined you know you can define a suitable mu for which satisfies those obedience constraints and so it any any any any any equilibrium of this game turns out to be a correlated equilibrium alright now can you tell me if the converse is true so this I did not get to discuss last time so yesterday so is the converse to suppose I gave you a correlated equilibrium and arbitrary correlated correlated equilibrium what is the corresponding communication system that that is there in the background so when when players choose a profile of reports are and a set of strat strategies like these deltas to map their messages to actions the payoff that player I gets from this profile is is equal to sum of m in m fine so when this is what player player I would get from this alright now this r and delta these are pure strategies so player can randomize over these and come up with a mixed strategy so mixed strategy for that would be a sigma i is a probability distribution in r i comma so I had refined this to be gamma gamma i right so gamma is the was the space of all r is comma delta is where r i is the report of player i and delta i mapped his messages to his actions ok so when I look at when I write sigma i of r i comma delta i this is the probability of choosing report r i and comma this and this strategy delta i ok alright and what we said was we can create a a correlated strategy like this mu which was just what is mu going to do think of it this way may who has to basically come up with it is eventually going to result in a probability distribution on the actions right so the chain that we have is that reports lead to messages messages lead to actions ok so for each pure strategy you have this chain and then you average over all the pure strategies because pure strategies are being chosen with us according to a certain distribution according to the mixed strategy like this so let us write it like this so first what are the messages that would lead to this particular action that is this for a particular report a report profile ok and then what is the probability of choosing this particular report and then what is the probability of choosing this function itself. So what is the what are the message you are something this is the probability of this is the probability mass of the messages that would lead to action x when the players have chosen a report profile r and a strategy profile this is not delta minus 1 this is delta inverse of this. So, delta inverse of x is basically just the m such that delta of m is equal to x and delta itself is so when I write delta of m I really mean delta 1 of m 1 all the way till delta n of m n. So, now you have this and you so now I need to multiply this by the probability of picking delta and r and that probability is simply the product over the players of sigma i delta sorry r i comma delta right and then sum over all such r i and delta r comma sum over all such r and delta this is therefore the final probability by which a certain action gets chosen. So, reports have been randomized over messages have been randomized over and the functions by that generate actions from messages the choice of such functions is also randomized over alright all of that has been averaged out and you get a certain probability of picking an action at the end of it all right and what we showed what we argued was that this is a correlated strategy. Now, what I am asking is what is the can do you have a converse. So, suppose I gave you an arbitrary correlated strategy can you come up with a communication system such that its equilibrium gives you this distribution in equilibrium the actions are being chosen with this probability one simple way of doing this there could be multiple ways. So, if you have some arbitrary mu what you could do is the following the channel just directly outputs the actions itself. So, the messages are the actions it ignores all reports there are no there is. So, it is independently of the reports it just chooses for you the actions as per mu and then what we are asking for is the identity function delta then which is just taking the action and just the suggested the action that is come in the message and just implementing that then is the identity function delta and ash equilibrium and it will be if mu is incentive if it satisfies the obedience constraint. So, if mu is a correlated equilibrium that means it satisfies the obedience constraints then the identity then all these deltas taken as identity which means by taken as identity means that. So, you take mu of. So, firstly you take the space of messages to be equal to the space of actions itself m equal to x all right. So, the messages are in fact the action that you are supposed to take. So, that is. So, when you get a message it is basically telling you take this action and now what you do is delta was supposed to map your message to an action you just apply an identity mapping. So, delta is the delta i of mi now is just doing mi or in short telling you what action itself is to be taken all right. And this if your correlated equilibrium is incentive has satisfies the incentive compatibility or the obedience constraints then such this will then be an such a strategy will be an equilibrium of this system with of this game. So, you have taken m equal to x and your communication system as I said has just ignores all reports there are no reports to be taken I mean you just do this effectively that is what happened effectively that is exactly what is happened. So, that the our communication system that is what I was saying that communication system noise is the playing the role of mediation. So, the communication system that you have built right that that you have trusted to generate or the random number generator or whatever it is serving the role of a mediator. So, instead of talking to each other we are we are sort of leaving the the coordination or correlation to that device right and that device has been built in such a way that you know it keeps this kind of confidentiality and so on and that actually helps us all of us do better. If everyone wanted to know what was going on what was happening in the lottery right you would actually achieve less they are not unique this is one way of doing it. So, the so what we wanted to show us the set of payoffs achievable in one is equal to the set of payoffs achievable in the other and so that is all I need to show. So, for me I just need to show that the distribution can be simulated once the distribution is simulatable then the set of payoffs becomes the same is the same. Is this clear? So, actually this is I find this to be very interesting I mean there are you can you know there are many many themes that one can build on top of this you know just any see essentially all any kind of decentralized control or decentralized decision making is finally about coordination fundamentally about coordination right and coordination when coordination furthermore with incomplete information each guy gets to see is a limited part of the underlying truth and yet we have to coordinate right. So, now that you can do this kind of you know you can a simple thing everyone thinks that they could do is just you know do this sort of coordinate by random doing a coin toss between each other's choices and that is your limited sort of direct communication option or public lottery or whatever. Once you do once you allow for once you allow a once you bring in this third party device right which everyone has agreed on actually you can actually do get achieve a larger set of payoffs through that. If you want to do this again think about this pre-play communication formally the essentially what every recommendation is doing is defining for you an information set it is telling you what possibly what are the possible things that could have been recommended to the other players based on that you just do a player just does a you know he finds a posterior distribution based on that given what has been recommended to him all right and then given that it is in his interest to it is in his interest to follow the recommendation that is your identity mapping being very the Nash equilibrium. Now in some places I have also seen a generalization of this essentially because I said you can also do the following you can look for a more general version in the following sense that well each guy has been you know the communication system is generating information for the players and now players are going to take actions right now you can so essentially down this is a this is a game of incomplete information between the players players have to now take choose their choose their strategies as a function of their information all right now you can look for any equilibrium in this doesn't have doesn't have to be this identity value equilibrium okay and some people call that as a correlated equilibrium all right now the revelation this this whole this whole thing that assume this that a correlated equilibrium simulates this simulates any outcome means that you can actually reduce that back down to the back down to a correlated equilibrium eventually you know where the action itself is being recommended and it has to be followed it is expected that that is what but so this is just a more general version of of putting the same same thing so so a correlated equilibrium of the game with communication was so I is just this mu such that this is greater than equal to and so and I said that this can be equivalently written as this x minus i in capital X minus i so for every recommendation it is in is in the player's interest to follow the recommendation than it so the I had mentioned something yesterday about the posterior also because you see and I just said posterior now the there's the reason this has been written without the posterior is because that is that possibility that a strategy that we are writing likes a recommended strategy Xi actually occurs with probability zero so you can't really divide by that probability and so exactly posterior is not defined so if you don't want to get into those technicalities what you you just you instead write it directly in terms of the joint distribution instead of getting into the posterior okay so the mu is written here that's why the mu has been written this way otherwise if you if these were all occurring with positive if the Xi's if each Xi was occurring with positive probability then I could just divide by that and then write a you know a proper conditional distribution of so probability distribution of X minus i you could write something like mu of X minus i given X i and so on okay alright okay let's keep this in mind and we will now write a more general version of this for games where where where players have some private information now I mentioned this yesterday the equilibrium of any communication system can be simulated by a correlated equilibrium today we argued that the converse is also true but remember the in a correlated equilibrium the communication is from the mediator or from that device to the players right so what has happened in this formula is that the reports have gotten averaged over so there is no role at all for the reports left even when I looked at the converse I just made it this independent of the reports right now why does what this effectively is saying is that there is nothing for you to report nothing for the players to report actually to the play to the mediator and the reason for that is players don't have any private information okay if players had private information of their own then the whole game would then the then there would be a reason for them to talk to the mediator and also listen from them to the mediator okay and also here what the mediator has to say okay so now that's what we can go through a generalization for this