 So, in the previous class we were, we talked about an example of pre-play communication and one of the, we actually looked at a paper of Robert Oman and we, what we saw there was that pre-play communication and or a pre-play non-binding agreement to play a particular equilibrium may not imply that the players will actually play that equilibrium, okay. So, then that effectively told us that Nash equilibrium there need not be self-enforcing, right. Now, what I will talk about today are models where an element of communication is involved during play, okay. So, communication during play, communication as part of gameplay. Now, what do I mean by communication as part of gameplay, this we need to be very, very clear about. Now, in any game obviously when there is a, when the game is dynamic, some information gets passed from one player to the next player, usually it is about the action of the player or something like that, okay. Now, that is not exactly what we mean by communication, okay. Now, what we mean by communication is communication is usually, what we mean by that is some, is some decision that a player takes which is not translating directly to its cost, okay. So, we make a distinction between two different things in games, one of what are called actions, the other are what are called signals, okay. So, decisions can be of two types, either actions or signals, okay. Now, there could be other types also which you can say, well, our contracts and so on, but we will come, I will come to that in a later state. Now, what is the distinction between actions and signals? The action, both of these are decisions, okay. Players take some decision in each of them. Action is that which appears directly in the cost, okay. It appears directly in the payoff. So, when you take an action, it actually incurs you a call. Signal does not appear directly in the payoff, at least in the simplest case. There could be cases where signaling is also costly, but that, let us leave that aside. Signals basically do not directly appear in the cost. So, what is the role of the signal if it does not affect the cost? The role of the signal is that it affects the information of the other player, alright. So, signals are decisions or some variable that a player conveys to the other player and that affects the information that the other player knows. So, through signals, a player affects the information of the other player. Through actions, also he affects the information of the other player, but he also affects his cost. So, a signal solely affects the information of the other player, alright. So, signals are therefore information carriers. And this is, these are actual actions that lead to impact the cost, alright. So, let us look at the simplest such game, which is what is called a sender receiver game, okay. And then from there we can take special cases of sender receiver games. So, sender receiver games have the following structure. There is a sender of information. He has access to some information, okay. Let us say for the moment let us just keep this, let us keep this simple. He has access to some information. Now, sender must send this information to a receiver, okay. So, this goes to a receiver. Now, receiver takes this information, uses this information to then decide what it must do, what the receiver should do, alright. So, the receiver is actually the one that takes an action, okay. So, the receiver takes an action, alright. And this action that the receiver takes determines the payoff of both sender and receiver, alright. So, this decides the cost payoff of both players. So, the sender does not have an action of its own, okay. Sender does not take any action of his own, because the sender can only do is control what the receiver gets to know, okay. Receiver on the other hand does not have any information of its own. The receiver only can take an action based on the information that it has received from the sender, okay. Let us take examples of this setting. And you will start seeing what many different variations that are possible. Now, in the economics literature, these are studied under what are called principal agent models, okay. And this goes into the domain of what is called contract theory. So, I can let us go over that for some examples from them. The simplest example is where there is no asymmetry of information, okay. So, in this case, so the simplest example, let us take game one, which is no full information. Now, what is this setting? The general setting is as follows. The general setting is that there are two players, a principal and an agent, okay. Players are principal among agent. The uninformed player, the player who does not have information is usually referred to as the principal, okay. So, this guy is your what we call the receiver. But in this case, there is going to be information will be trivial. But nonetheless, this is we can liken this guy to be the receiver. And this fellow is your sender, okay. So, this is essentially the principal is the uninformed player, uninformed player. This is an extreme case, obviously he could be less informed than the agent and all that stuff can be incorporated if necessary. The agent is the informed player, okay. Now what is the idea? The idea is here, the way the model in which we are going to study this is where the principal is committing first, okay. The principal commits its strategy first. Now, the way we should think about this is that there is a principal who is let us say a manager, okay. And the agent is a worker. The manager wants to get work done out of a worker, okay. But and so what it does is what the manager does is he announces a reward scheme. And if you do this much work, then if you put in this much effort, this is the amount I will reward you with and so on, okay. Now the nature of what the reward scheme is etc. is basically becomes the subject of or it gives you different types of models, alright. Now essentially the problem therefore is that the principal has wants to get a certain task done and it must delegate this task to an agent, okay. Because it cannot directly observe the you know the situation the environment needed to achieve that task, right. So the agent might be so if you want to take a cyber physical system type of context then the principal could be a some sort of infusion center and the agent could be sensors that are actually observing that are actually sensing the field. They are the ones that get the information. They need to send this information over to another to a higher level aggregation of center or a fusion center where this where this aggregation will be performed, okay. So the point is the problem for the principal is okay how should I be rewarding the agent in order to achieve the get this work done, okay. So let us take the simplest sort of case. What the principal will principal does is let us say offers a contract, principal offers a wage W which is a function of the effort E. So this is your wage W and this is the function of the effort E made by the agent. So the this is the order of play. So the principal offers a wage which is going to be a function of the effort that the agent that the agent will be. So what the principal is offering is actually a function, okay. The principal strategy this W is his strategy it is a function. The function which says that if you make so much effort I will give you so much money if you make so much effort I will give money etc. So okay. Now the agent has the because this is a this is a contract type of setting there is an additional condition that the agent has the in the economics literature they take what is called they put in this condition of what is called participation. Essentially the agent has the option to accept or reject the offer, okay. And if he accepts then he has to choose an optimal effort level or E star. So in the case where there is full information what is why is this a case of full information. The reason this is a case of full information is because the principal is offering a wage as a function of the effort that the agent makes. And the assumption here is that the principal can observe the actual effort that is being made. So let we will come to this in little more detail but let us think about each of these things one by one. Now suppose so let us suppose the agent makes an effort E, agent effort is E okay. And then from there that gives you know the monetary value of that to be to the principal the value of this effort to the principal is let us say Q of E. This is what the principal gets this is the value to the principal. So what the principal is the value the payoff that the principal receives the principal is a function of the effort that the value that it has received from the effort okay minus whatever it has paid the agent for the effort okay. So it has paid W of E for the effort okay. So it is a function of this alright. So this is the net payment it has received Q of E and it has paid W of E the net is the F for Q of E minus W of E and V of that is it is a function of that is it is it is the payoff that the principal receives. Now as far as the agent is concerned remember agent has two choices E for either accept or reject and if he accepts then he can talk of okay how much is the effort that he makes right. So what we can think of the way we can think of this is that the agent for the agent to make an to accept he has what is what we can say is a reservation level utility. So if he needs the utility at least of a certain level in order for him to accept accept the contract okay. So it is some base some bare minimum utility that he needs otherwise the otherwise it is the you know getting into this contract is not worth it okay. So it is some threshold level this could be derived from some whatever what the competitors are paying or whatever does not matter but some reservation level utility that he must get and this the agent gets agent utility is a function U of the effort he makes and the wage he receives okay and usually it is assumed that this in U of E comma W this is assumed that this decreases with E okay more effort means less utility and increases with W alright. So the agent utility is this and we assume that there is a reservation utility that means he must he needs some basic level of utility otherwise the whole thing is not worth it. So you so he needs U of E comma W of E to be greater than equal to some U bar where U bar is a reservation utility. So now if you think about this problem now tell me how should the agent play maximize what maximize maximize this okay okay alright you maximize this okay then alright fine we so the agent maximizes you alright then how does the principal play so let us let us ask okay what is the what is the what should be the how should he find what should W the optimal W satisfy yes what else this is of course it must satisfy this reservation condition or what is called this is also called the participation constraint. So let me ask you this suppose there was no participation constraint okay suppose the agent had to participate okay the agent had no choice he had to accept the contract and what would be what would the principal pay him 0 would in fact be minus infinity. So the reason this condition is put here because it is it you have to prevent slavery if without this essentially what in fact minus infinity is a solution you can pay him minus infinity he has no choice you just keep him make some effort right whatever effort yeah but minus infinity gives you infinity as payoff. So you basically take all you take all his money essentially that those are separate that is a separate matter even if it is bounded below you get that bound but that is the point right so essentially without the participation constraint this problem is actually ill-posed because it effectively mean that you can extract as much money as you want from an agent because he is bound to you can give there is a there is a trivial there is a trivial maximizing contract in which the agent pays you to work to make him work no no no whatever it is I mean it is there is always this trivial contract right because we so we will be V is the payoff it will keep increasing with with with this with this argument so I can keep making W as large as I want no even if it is even if it does the point is if see the if you make V W W can be made as large as you want the matter is over I can even make W a constant minus infinity you are charged minus infinity to be in this to be to do even if you do 0 effort you have to still pay me minus infinity right so this so without this that is the answer is actually trivial ok. Now now you can ask yourself ok well then if there is if this is the important constraint here then watch how should this how should the how should the W be determined well you so the principle will not pay will not may cannot charge minus infinity but what he what would he do then well he will charge him the bare minimum to make sure that he is in the contract he accepts the contract right so so the W W star the optimal wage would be such that this is always equal to identically right so effectively you can solve for this this this equation here and get W star as a function of it ok you want this to be identically equal to equal to you by it is the bare it is the minimum value of it is the minimum wage such minimum wage as a function of the effort that makes this equal yeah for all because we are allowing it to be a function of it all right now having done this essentially the what has happened is that the agent has nothing to optimize because his utility is now constant he is always going to get the U bar itself right because the wage has been fixed in such a way that U of U of this is equal to equal to U bar there is nothing for him to optimize so any effort is any effort gives him the same amount of utility right ok so so then effectively what what we can say is well what the maximum value that or payoff that the principle can receive is then maximum of of this because they all the efforts are give the agent the same utility so the maximum that the age that the principle can receive is this ok now there are more more variations one can make on this and you can dive deeper into this but that is that is not the that is not the direction I want to go in the thing I want you to appreciate here is see what is the way this contract is based ok the what is the information that the principle has the information that the principle has is that he can actually observe the effort that the agent makes ok which means so suppose this this agent has been told to go and you know survey survey a certain area ok it is some drone you have to survey a certain area you can actually observe with that the he is gone to this is gone to that etc etc now you may think this is some kind of a silly thing but it is actually a very very important thing you know today in fact sales agents that come to your home you know maybe LIC or you know whatever sales agents that come go from door to door they have to keep their they have they have usually an app installed and they have to keep that app on because they are the company monitors whether these guys are actually going to the in fact to the said locations or not ok they have to keep that app on and it keep reporting GPS location that is what because they want to the company wants to monitor if these guys have in fact visited these locations. The reason this because this is becomes important is because now what happens if you if the principle cannot observe E ok so the effort that the agent makes is not observable is not observable by the principle then you start getting into all sorts of problems yeah yeah yeah this is this is this is the best he can get yeah nothing that is why this is this is a kind of degenerate problem because of this reason because no so firstly we are we are in a sequential setting ok so we have to so we will have to solve this in a stackable bug sense so there is a declared contract and then there is a response yeah yeah so this is yes so that is that is also true ok so it is true that the agent can make essentially any effort and then that is that is that right so that is so that is also true so this is this is the maximum he can get it is also the minimum he can get you can also find sorry you can also compute some correspondingly the minimum that he can get ok alright so now let us look at a different version of the of this thing now let us suppose you have another another case where the where the principle cannot observe the effort ok now so what are the two things that the principle could observe here one here we are assuming that the principle could observe the effort alright the what is the other thing that the principle could observe that is known but as a during gameplay what is what can you observe so here is here is here is a here is a possible possible alternative think about the objective of the principle the principle cares about q of e minus w of e ok w is what he decides q of e is the value that he gets out of the effort now you might ask ok why does the principle want to know what the actual effort is what he cares about is the value right so long as the principle is so long as the agent is going and doing enough sales for you why do you care if he is gone to this house or the undone how much spent time he spent talking to the customer whether he is visited every house etcetera etcetera what matters is that he is gotten he is sold enough items right so q of e is the value that of the effort that that the principle gets out of the efforts of the agent principle does not get money from effort itself does not give him money what he get effort results in this in q of e and that results in benefit to the so you may say well then another approach is that I can that the principle could base is the wage on what on q of e rather than ok so that is another type of model so there now the contract is not a function of of the effort because not a function of the effort but a function of the output ok so that results in another type of problem ok but let us take of extreme case first before we come to that let us let us take an extreme case where the principle can observe neither effort nor output ok.