 So, in the previous class, we talked about problems where players were allowed to get into binding contracts with each other and we asked what kind of payoffs that can players achieve or that are attainable by the players under such kind of contracts. So, we were not talking about designing the optimal contracts, but rather asking if players have to get into binding, have the option of getting into binding contracts and arbitrary contracts are allowed, what kind of contracts would they sign and with such contracts being added and augmented to the game, we wanted to know what kind of payoffs can be achieved. So, today what we will do is we will relax the condition that players are allowed to get into binding contracts and we will however still allow players to communicate with each other. So, there will be communication and players will have a wide amount of communication possibilities, but they will not be allowed to get into binding contract. So, that does not mean that binding contracts are impossible, it just means that every contract need not be binding. So, now in order to talk about this, let us consider the following, we will consider the following game as an example. We have a game with two players, player 1, player 2, player 1 has two strategies X1, Y1, player 2 has two strategies X2, Y2, payoffs of the players are 5 comma 1 here, 1 comma 5 here, 4 comma 4 here and 0 comma 0 here. Now, there are three Nash equilibria in this game, you can eyeball them quite easily. The first Nash equilibrium is X1, X2, there is another Nash equilibrium which is Y1, Y1, Y2 and in addition to this, there is also a mixed Nash equilibrium in which player 1 randomizes say half X1 comma half Y1, half X1 comma half Y1 for player 1 and half X2 comma half Y2 for player 2 and the payoffs that they get are, you get 5 comma 1 here, 1 comma 5 here, you can check that they get 2.5 comma 2.5 in this. So, the problem that we now, we are now going to address is where players are allowed to communicate but there is no scope for them to get into binding contracts. So, the question therefore for us is what does it mean, what kind of strategies are now feasible now that players cannot get into, can communicate but cannot get into binding contracts. So, in the absence of communication players were randomizing independently. So, each player was choosing his or her mixed strategy independently. So, and from there we concluded these three as being the Nash equilibria of this game. However, now that communication is allowed, players need not randomize independently means players could potentially do enter into correlated randomizations. All right. So, in particular they can in fact play what is any correlated strategy. So, any correlated strategy you in this, I am using the notation from the previous class. So, any correlated strategy could be used but remember when players say well let us play this correlated strategy what does it mean for example, it would say let us say it could a one correlated strategy would be there are these four possible combinations here. If a correlated strategy could say well we will play each of them with probability 1, 4. So, x 1, x 2 with probability 1, 4 x 1, y 2 with probability 1, 4 etc. Now, or it could be that players could say well let us play x 1, x 2 with probability half or y 1, x 2 with probability half that is another correlated strategy or another possibilities let us play x 1, x 2 with probability half and y 1, y 2 with probability half etc. Now, remember any correlated strategy that they just discuss and talk about is not binding on the players. So, they may say okay let us play this but when they actually go to choose a particular strategy they may not actually stick to that stick to whatever correlated strategy is part of the plan. So, what kind of strategies can actually then be are actually feasible under the fact that players can communicate but there is no binding agreements. So, or rather let us say forget what kind can you give me an example of a strategy that is feasible under binding agreement but without under communication but with no binding agreements. So, you cannot just pick an arbitrarily correlated strategy because that is you know that actually is because there is nothing to bind the player to playing that particular strategy right. They were also see remember let me also point out one more thing. The set of strategies that can be implemented under correlation under communication but no binding contracts. Now, that set of strategies okay how is that related to the set of strategies that were that are implementable with binding contract with communication and binding contract which one is the subset of which one of course one is a subset of the other that is why I am asking this question but see the point is see think about it this way see if a strategy is of the kind that a player would not like to deviate from it even though it is not binding then he would not like to deviate from it even if it is binding right. So, this the set of payoffs that players can achieve with binding contracts is actually larger. So, the point is that once the once players are allowed to get into binding contracts essentially that that widens the scope of the scope of things that they can do. So, in fact the kind of games that will result out of that are the kind of games that we just wrote in the previous class where there is one extra contract and that there is an extra decision then whether to sign is that contract or not right. So, contracts are the so you so what you have is a game that you begin with and then you extend that game with additional options that come from the contract and you ask which contract would you want to sign any of these contracts alright knowing that when you sign it is binding upon you whatever you are signing on alright ok. So, now that so the so the so now with binding contracts out of the question what play what can players achieve with communication. So, here I have to also make precise what we mean by communication ok. So, again once again you can go in two different ways here one you can start modeling the operations of communication where one player says something then the other player responds or third player says some then the first player plays response to this response etc etc that is effectively communication alright or we say well we do not bother about who is saying what we just say what is the result of all of this communication in the sense that in the in the sense that after all of this communication what is the action that you are going to be chosen ok. So, this is what is this way of modeling what it does is it makes communication implicit essentially we are not modeling the actual exchange of signals but we are saying what is the final result of those signals that are exchanged alright. So, now here so here what we are talking about is communication that is happening between in which players are openly then directly communicating with each other directly communicating which I will explain what indirect communication also means in the moment. So, what this means is that players can conduct a joint experiment like a coin toss or whatever all observe the outcome of that experiment and then decide what they should be playing based on an outcome of the experiment which all of them observe alright. So, this is direct communication essentially so they can this is what it would mean to correlate their actions you know by this is what it would mean to correlate their actions alright. So, now what do I mean by this so I will give you an example of a strategy that players can create now with communication any correlated strategy could be used but not every correlated strategy is something that they would want to stick to. So, I will give you an example so here is an example of a strategy that they would want to actually stick to. So, for example they may say let us play the first equilibrium x1, x2 so these are three Nash equilibrium alright. So, you take the first equilibrium x1 play that with probability half and play y1, y2 with probability half. So, what does this mean how are we implementing this players toss a coin everyone witnesses the outcome and everyone has agreed that this is what will be played with based on the outcome. So, if it is heads you will play they would both play x1, x2 if it is tails they would play y1, y2 alright. The question so you can I can this now is another option that is present in the that is now present in the game. Now question is with this option now player has to go back and but before the coin is tossed decide whether he actually sticks to this option or not and so what are we assuming when he sticks to this option there is no this is not a contract. So, what we are only thing what is being assumed is that if a player will now can only unilaterally deviate from this alright. So, this they have come up with this particular distribution and now they can when it when it comes to actually playing they can only unilaterally deviate from this alright when it comes to actually pressing the button they can unilaterally only unilaterally deviate. So, what does this mean for example if player 1 decides to deviate from this he can say let me play a pure strategy x1 or let me play a pure strategy x2 or let me play something else altogether. But he is when he is unilaterally deviating player 2 is still assumed to continue to play according to this experiment that means if it is heads then he is going to play his portion of the strategy which is x2 and if it is tails he is going to play his portion of the strategy which is y2 alright. So, the question for each player is now assuming that the other guy is sticking to this correlated strategy would I want to stick to this okay. So, now if you look at this particular this particular strategy that has been created is this something that players would want to stick to assuming the other guy also sticks to it yeah. So, this is actually a combination of two pure strategy Nash equilibria of the game. So, it has obtained by taking half of this Nash equilibrium and half of this Nash equilibrium. Now, if player 1 suppose deviates from x1 to x instead of playing this suppose he decides to play x1 okay let us take a pure strategy x1 right then what would happen because it is a Nash equilibrium he of course here it remains x1 x2 in this one but here shifting to x1 is actually not profitable because y1 was the best response to y2 because y1 y2 is a Nash equilibrium. Suppose player 1 plays x1 as a deviation from this half x1 x2 half y1 y2 this one okay p2 and p2 continues to play as per let us call this mu as per mu okay. So, that means he is going to play so if since he is playing as per mu what this means is that if it is heads he is going to play x2 otherwise he is going to play y2 which means that essentially he is going to play x2 with probability half and y2 with probability half right. So, player 1 suppose he deviates to a let us say to playing x1 then look at the payoff that player 1 is going to get player 1 that will get a payoff x1 comma whatever it is that comes from whatever it is that player 2 is going to do. So, he is going to do x2 with probability half comma y2 with probability half. So, this is the payoff that player this is the deviation payoff to player 1 deviation utility to player 1. So, player 1 has deviated from his from playing x2 in this in this coin in this in the tails event instead of playing x2 he is now saying he is going to play x1 right. So, then in that not x2 what was I saying y1 instead of saying playing y1 he is going to play x1 right, but then y1 was a Nash equilibrium was part of a Nash equilibrium. So, what this means is this this here is actually less than equal to and the same thing would hold if I said if player 1 instead of playing x1 as a deviation was to suppose playing y1 as a deviation if he instead played if I replace this thing by y1 here I would get y1 here out here and this would remain I would this would remain y1 I would get y1 here sorry not here I would get y1 here in place of this x1. Now the half here is not important at all actually that this is in fact 50-50 probability what is important is that these are convex combinations of the Nash equilibria of the game if there are more Nash equilibria you can put all of them in and you take all the pure strategy Nash equilibria and take their convex combination in fact you can take even mixed value strategy Nash equilibria take their convex combination there will be no incentive for any player to deviate because whenever he deviates he is basically going all these inequalities are only pointing in the in one direction he is going worse off than in each of these terms right and you can deviate to a mixed strategy or whatever does not matter. So, which means then with this kind of direct communication where the mechanism is a joint experiment is conducted everyone witnesses the outcome of the experiment and and and based on that outcome the the strategy this thing is chosen the the strategy is chosen this is this kind of this kind of in this sort of communication the payoffs that can be achieved are is actually the convex hull of this set of Nash equilibria of the game. So, the pay so payoffs that what is what do you mean by a converse so it is in fact so that is the exact set of this thing so it is so that you have the converse here in that sense there is a converse payoffs achievable okay so now these so once again I will emphasize this direct communication as an aspect because we will now move to indirect communication okay so once again what is what are we assuming about the direct communication about communication here is that the strategy is designed in such a way that it requires all players to witness the outcome of the experiment okay and then based on the outcome they have to take the choose their choose their actions all right now the so observe now you may think that okay what is the big deal about observing the outcome of the outcome of the experiment but it turns out that if players are allowed to communicate indirectly then they can actually do more all right so let us actually calculate the payoff for just for for completeness sake so what is the payoff that they can get here in this example they can 3 and 3 no half of half this and half this they can get 3 and 3 right so the payoff that they can get is this leads to the payoff 3 comma 3 for the players now can players do better than 3 comma 3 and that is what I will show you that it is actually possible to do better than 3 comma 3 okay by by allowing mediation I mean you can the half is not important as I said it is like you can take any convex from yeah yeah yeah because each of these terms right does an inequality oh you mean this you mean it would not be 3 comma 3 anymore right yeah yeah yeah yeah so but I this is I will just give you an example I mean this is so so actually we I will eventually we will draw a figure in which all of the and all the regions are indicated so that that is the the 3 is not that important that is actually the highest the better payoff that they can get okay so because of symmetry that you will you will see that they can in fact get do better through indirect communication it would still be it would be still achievable under this under this kind of communication under direct communication no problem yeah that would still be achievable under under direct communication so the set of payoffs achievable through direct under direct communication is the convex hull of all all the set of all Nash equilibria of the loop okay okay so now here is just now as I said this this requires them to witness the experiment witness the outcome of the experiment now if you let us now generalize this a little bit and what I will do is I will allow for mediation okay now mediation may look like initially may look like I am allow bringing in one more player you know mediator into the picture but what you will see is actually that there is that mediation is in fact just another form of indirect communication between the players okay so this was direct communication where players are where all you know players are witnessing everything together you can once you allow for indirect communication you have you you are basically you have the you can actually achieve more all right so now let us let us bring in so first let us bring in mediation so how does mediation how would mediation work so suppose you have a mediator okay now what the mediator would do is mediator would announce announces that he is going to be using a correlated strategy okay mediator announces a correlated strategy strategy mu which is a probability distribution on the set of all action but now here is what he would do the mediator only communicates confidentially with each player okay there is communication only there is no common experiment that he can that all players witness he communicates confidentially with each player means that he will tell each player a recommended action right so the mediator so he for example he will tell player 1 you play x1 he will also tell player 2 confidentially that you play y2 okay players would know that that the recommendations are being generated from a certain correlated strategy this correlated strategy mu all right but they do not know what has been recommended to the other player they just know that the recommendations have been generated through this they know their own recommendation all right but they do not know what has been recommended to the other player this is clear this is the role of a mediator this is indirect communication okay so here is an example so suppose a mediator recommends this so with probability one third he says you play your first Nash equilibrium x1 x2 with probability one third you play y I am going to give you y1 y2 and with probability one third remember there was this another payoff here 4 4 payoff yeah so with another probability one third I am going to recommend you y1 x2 the as I said the mode of communication is this that you have a mediator who talks confidentially to the players okay so this is this is your echo to player 1 this is a echo to player 2 each player only hears his own record he knows that and each player knows that the recommendation has been generated through this probability distribution mu is this clear okay now what this does is basically what is happening behind the scenes is not known to the players players do not get to know what happened in the experiment they just know what you they they just know that what part they need to play respectively right they just know that okay mediator has told me to play this I will that that is what I need to and that is what has been that is all he knows the player knows okay he does not know you know whether tales came whether heads came you know whatever whether eyes game or whatever yeah yes yes yes yeah so this is a trusted mediator okay so the mean you will later see that the mediator is actually just is not really another player it is just a it is just a sort of a thought device okay so eventually what we will do is we will reduce this to a game of indirect communication between the players itself okay so so for the moments are so that is why I assume that the you know mediator is some some software that they have that is generating these recommendations and they have all trust they all trust that software so the framework is like this that the mediator just recommends something to these individual players and now the because players do not know what is happening behind the scenes right they each have partial information about what has actually happened okay they only know what has been recommended to them okay so I will give you an example you know a lot of platforms we use today are actually of this nature right so when Google Maps tells you that you know take this route you just know that you Google Maps has told you to take this route maybe there is another person who wants to go from the same origin to the same destination and Google map has said you know take some other route okay or let us say in finance broker tells you you know buy this who knows maybe to some other client he is telling is telling something else he is saying do not buy this buy the or buy that buy something else or he may be saying he will sell this right so all of these actually form all of these platforms that we use are actually what they are actually doing is doing this kind of mediated communication we only know what has been recommended we have a general impression of what kind what you know how the recommendations are being generated now if the platform is actually doing what it should be doing okay then it should it should be some if the platform for the platform to let us say be successful what do what do you expect to have what should how should it produce its recommendations so for example why would I want to keep you know why would I want to keep using Google Maps or rather why would I want to follow the route given by Google Maps so it should if I deviate from Google what has been recommended to me by Google Maps it should give me a higher travel time right I do not care what has been recommended to the others what I only know is somehow Google Maps is figuring out its set of recommendations in such a way that if I stick to my each if I if each guy sticks to his recommendation right then the rather for each guy it is optimal to stick to his own recommendation assuming the other guys stick to it okay so I will come to this remember this point so the essentially what is what is going on with a mediator is some sort of this sort of partial information sharing need to know basis you each player is just told what his own recommendation is and based on that he needs to now he needs to now take an action so but for a strategy to be implementable through a mediation like this what should satisfy it should satisfy a basic obedience requirement that it should be it should be in my interest to obey the recommendations given by Google Maps assuming the others also are obeying all right so let us come to we will come to this in a bit more detail so here is then the strategy that this mediator announces okay now what does this mean let us let us understand this a little so so let us see how this strategy would work so suppose the mediator says x1 is the recommendation to player 1 now what does player 1 know from this recommendation he knows that player 2 has been recommended x2 with probability 1 since he has been player 1 has been recommended x1 he knows that this mediators mediators coin toss has resulted in this right because that is the only way he gets to he gets a recommendation x1 so he knows that player 2 has been recommended x2 now if player 2 is playing x2 is it optimal for player 1 to continue to play x1 yes all right so if player 1 here is x1 it means that implies that player 2 has been recommended x2 so as I said this player this mediator forgot to write this maintains partial information okay now suppose the player 1 hears y1 as a recommendation now if he hears y1 what has what does that mean what can he say if he hears y1 then he means that either this has come about or this has come about but because the players do not get to witness the experiment itself he does not know which one he just knows that one of these two has come about but from there what he can say is that well player player 2 has been recommended y2 with probability half or it is that he has been recommended x2 with probability half okay so there is a 50 percent chance that he is that player 2 has been recommended y2 and 50 percent chance that he has been recommended x2 that player 2 has been recommended x2 okay then player 1 knows that 2 has been has been recommended y2 with probability half or x2 with probability half right now look at the payoff now now suppose if you if player 2 is then going to be recommended x2 with probability half and y2 with probability half then what is the then you have half of this plus half of this as the payoff that player 1 sees right so so what is the payoff he is going to get from x1 he is going to get 2.5 what is the payoff he is going to get from y1 he is also going to get 2.5 so both x1 and y1 given are giving him 2.5 he is been recommended y1 so it is in his interest to to play y1 okay so in other words he has no rather let us put it this way he has no incentive to deviate from the recommendation okay so no incentive to deviate from the recommendation and a symmetric thing holds for player 2 as well now with this now what is the payoff that they can get can you calculate this is of course again symmetric yeah they get 3 3 and one third so earlier they were getting 3 3 as the symmetric one this is they are get they get 3 and one third okay so the so what has happened here the the way this has worked is this mediator has maintained a kind of confidentiality need to know whatever you want to call it and through that implemented this strategy now I will tell you you can see why confidentiality is actually important here so suppose okay suppose now here if you look at what player the case with player 2 okay player 2 has been recommended x2 here okay he is also been recommended x2 here now if player 1 knows okay suppose this was the outcome okay let us say this this thing was the outcome then player 1 would get outcome of the experiment of the coin toss mediator tosses the coin it turns out to be this this is the outcome so he will tell player 1 y1 and he will tell player 2 y2 now when player if the outcome of the coin tosses revealed to the player then player 1 when he sees y1 he knows that player 2 has been recommended x2 if the outcome is also revealed then he knows that he is been recommended x1 and player 2 has been recommended sorry he has been recommended y1 and player 2 has been recommended x2 okay now if player 2 has been recommended x2 is y1 the best response to that no y1 is not the the best response, X1 is the best response. X1 actually is strictly better than Y1. So player 1 sticks to playing Y1 only because he is not been told what the outcome of the coin toss actually is. It is because player 1 is just knows this. It could be a Y2, it could be this with probability half or this with probability half. And as a result of that player 1 agrees to play Y1. If there is this too much information then player 1 would not agree to play Y1 in this case. So this is where mediated versus unmediated makes a difference. See the direct communication that I was telling you is essentially where the players can observe the experiment itself. Whereas here there is no observation of the experiment. They just observe the action that has been recommended to them. And there is a vast difference between the two. So here basically the mediator is saying Tumam Khau, Gutliy Ahmad Ginoz do not worry about what is happening behind the scenes. You have been told is just do that. So this is obviously a much more general strategy because now there is also a possibility of partial information which is built in as a part of the definition of the strategy. So with direct unmediated communication such a strategy cannot possibly be played because players have to witness the experiment when they are communicating directly. So you ask me what is the definition of direct communication. Direct communication is where the sample is actually directly observed. Whereas here function of the sample is observed which is the recommended action. Is this clear? So now let us come back to the Google Maps example. So Google Maps is doing something like this. You know Google Maps, Zomato all of these platforms they are telling us you know eat this, eat that whatever, go take this route. They are suggesting other articles etc. etc. What all of them are doing is something like that. If Zomato tells everybody to go and eat in a particular restaurant, everyone will go there and crowd the restaurant and nobody will actually get a great experience. So the kind of optimal thing for Zomato to do is actually spread around its recommendations also. Thinking of various things and so on. So actually then this kind of puts into this thing. If you think of mediator not as a fictitious entity but in fact as a real player then you can actually think of things like what are the feasible recommendations, what recommendations maximized if mediators revenue, the platform revenue etc. etc. All bunch of other kind of questions can be thought about in this.