 So, when we started talking about non-cooperative games what was our main what was the assumption we said well non-cooperative games are games in which players cannot communicate with each other and have no scope to get into binding agreements right. So, now the extreme of this is where both communication is allowed and binding agreements are allowed alright. And we will start from there and we will starts allowing players to communicate you know in a graded manner and sort of we will start with the simplest case where both communication and binding agreements are allowed ok. And then we will build up from there and eventually get to we will come to you know we will come to games we will come to certain games where you will see results like moral hazard and adverse selection and so on actually come out as core always ok. Now, if you want to so the first setting we are going to consider is where players can get into binding agreements. So, there is some mechanism for which says that players which will enforce the agreement if players say that if players sign the agreement it will get enforced ok they cannot sign and then say well like I am not I am not going to play ok. So, there are two types of questions you can ask here first is what kind of agreement can be reached alright. What is an agreement point that can be reached alright and that goes into what is called bargaining theory. So, where players have to come up with an agreement between themselves essentially bargain to get to an agreement between themselves alright considering their utilities and so on. Now, so what the thing that I am going to talk about is not what kind of agreement, but a problem of signing whether to sign an agreement or not given when presented an agreement alright. So, so here is the here is the setup so let us so these are what are called games with contracts here as I said we have allowing the most extreme form of communication arbitrary communication and binding contracts ok and as I said we are not worried about not designing the not designing the agreement not designing the agreement that is not what we are concerned about, but whether but asking whether to sign whether sign it or not and now for a non-cooperative game what would we see we said we should analyze this based on the Nash equilibrium because and the reason for that taking that solution concept because is because that was that is what made sense in the absence of any communication right. Now that we are allowing communication and binding agreements we have to change our solution concept also we cannot continue to stick to the Nash equilibrium solution concept anymore right. So, the solution concept that we will want to look for is we want to characterize the set of all payoffs achievable through any contracts that can be signed by any subset of players. So, look at how sweeping this is any contract that can be signed by any subset of players alright. Now this looks extremely ambitious actually to think about it so essentially just imagine this so if I wanted to I am thinking of a game let us say you have a static game to begin with alright. So, let us say let us actually write one out let us consider this game with two players so this is player 1 this is player 2, player 1 has two strategies X1, Y1 player 2 has two strategies X2, Y2 players are maximizing the pool these are and the payoffs are this. So, this is basically we can I will write out the payoffs can someone identify what this game is no no it is not dear ever this is this is prisoner's dilemma right. So, if both players stay silent they would have gotten 2-2 if but if you know if one is staying silent and the other this thing the other testifies you the one who testifies would get 6. So, this is maximizing so 2 6 is better than 2 right so but so as a result of that 1-1 is this so you can see this is this strategy dominates the other 2 alright the other strategy alright. So, so this is essentially just prisoner's dilemma and it has this is its equilibrium ok. Now, if you want to allow players to communicate so this is we are now allowing the prisoners to communicate right now how do I model something like this this is this is it is incredible that some theory can even be built which allows for so much here. So, if I wanted to model communication for example, what would I have to do each player would say a word the other player would hear it maybe say another word in return and then the other player would say another word in return and so on. So, then how many actions do you have for at each such around every word in the dictionary is a potential action right or every possible sentence is a potential action. So, you can so you can see the you just modeling this sort of communication itself leads to a a tremendous blow up in complexity and there is no possible way by which we can hope to analyze something like this. So, that is why I said it is incredible that something actually can be said about games like this where players can actually communicate and get into contract ok. So, so what we will do is we will not model the communication itself, but the result of the communication ok. So, that this is one of the enduring themes this is this is the sort of way of of approaching games was developed by Roger Myerson it is an enduring theme it leads to very elegant formulations he again Myerson again got a Nobel Prize in 2007 or 2008 date about. So, it is a very nice way of thinking about it and you will realize that a lot of you know the kind of problems that we encounter in you know in say for example, consensus coordination problems flocking and all of these a lot of them can be framed you know using this kind of framework that Myerson has ok. So, so what you do is do not model what actually is exchanged between the agents, but what is the result of it all right now here is the idea. So, this game as I said has only one equilibrium now let us allow for a following contract ok. Now, the contract says the following that we are going to allow this contract which is that if both players sign the contract then they agree to play X1 comma X2 ok. So, if both players are going if both players sign the contract then they will play X1 comma X2. So, X1 X2 remember was this silent silent type of outcome right. So, if both players agree to play X1 X2 sign the contract then they will they agree to play X1 X2 and there is a mechanism to enforce this all right. And if it is if the contract is signed by only one player, one player then he or she will play plays Y1 or Y1 or Y2 based on who is signing all right. So, what the choice for the players is whether to sign this contract or not all right. So, what you have is you have started off with an original game of prisoner's dilemma and now there is this some lawyer or someone who has come in between and says that well I am going to give you this contract you have this option ok to sign this contract. If you sign it if both of you sign it you are bound to play X1 and X2 if only one signs it the one who is signing it will is saying that I am going to play Y1 Y1 or stroke Y2 ok. Now, what this does is it basically now expands the game. So, the players now have now have three choices the player one had early his earlier choices X1 Y1, but he also has a third choice which is to whether to sign the contract or not all right. So, let us call that choice as one. So, if it is likewise player 2 also X2 Y2 and S2 the first part of this matrix is the same as before. So, you have 2 2 0 6 6 0 and 1 1 and now what does it what does this say so S1 is the action of signing the contract. So, if player 1 signs, but player 2 does not sign then player 1 is basically going to play what is the contract saying he is going it is he is then going to play Y1 all right. So, he is going to then this is what happens to. If likewise if player 2 signs, but player 1 does not sign then you get this and if both sign then they are both bound to play X1 X2 and so the payoff is going to be this all right. So, what have we done we have removed all the communication in that could have occurred and directly modeled the outcome of the contract right what is the action that is that is going to eventually result from this from all this deliberation. So, remember that is what this says right if so if player 1 signs the if both players sign the contract then they will both play X1 X2 and so the payoff that is going to result is 2 comma 2. So, if form if player 1 plays S1 and player 2 plays S2 this is what is going to be the payoff is this clear. So, this is this is the action of signing this contract yeah still maximum all right and the reason you saw you these 2 rows are repeated and this column and this column are repeated is because if only one of them signs then the commitment is to play the one who signs is committed is committing to play Y1 or Y2 player 1 signs then he is committing to play Y1 if player 2 signs he is committing to play Y2. Now so then effectively now with this lawyer now that the lawyer has appeared and you have this contract also the problem choice for the players is to whether or not to sign this contract. So, what the resulting game is now this game with 3 strategies for each player or 3 actions for each player yeah what is the equilibrium of this both sign is the equilibrium right. So, this is the equilibrium now. So, it turns out then that this is your equilibrium all right. So, what have we assumed remember we have assumed that when you sign you are bound by whatever the contracts ask you to do if you are not signing then you can do whatever you want right. So, which means that there is a mechanism to actually ensure that whatever they sign on is in fact happens all right and in that case now with this contract in the picture you get a new equilibrium emerges in which both players are okay with signing the contract and then that gives them 2 comma 2 all right. So, yeah no that is so part of designing the contract is actually this what should be the you know what is the sort of there is a name for this I think it force major what happens if you know one of the parties that I just from the contract okay. So, when you write the legal documents like when I work with industries and all that we have a there is a section in that where we have to put in this that suppose say what happens for example if the if for whatever reason one of the parties breaks away from the contract then what would the other party do. So, we agree to what would happen if we disagree in a to a partial extent if we completely disagree both of us not sign then what happens is also a different that is a different issue altogether in which case we know the contract does not exist. But if only one of them signs then what happens. So, this so it is part of design I mean the whole contract must encode encompass all of this for it to be a well defined contract no not necessarily now. So, what is happened is so so so the situation that has a reason is this communication has allowed has been allowed this option has come up there is a mechanism to enforce this okay right now all I am saying is that this option is available we will later generalize and say let us suppose all such contracts any any and all you know or an arbitrary set of contracts is available then what happens okay we have not got there yet but with this contract this is what we can say is this clear okay alright. So, now suppose let us generalize this suppose there is one more contract okay. So, another contract here so here is another contract again if both players sign then they will toss a coin and choose X1, Y2 if heads and Y1, X2 if tails I will just write out the full matrix. So, there is now this new hat contract and I will just pull out the full write out the full matrix then you will see the water nature of the contract is. So, this portion of the matrix is same as before okay. So, now let us understand this where where all these terms are coming from. So, now here S1 hat is the act that player 1 signs this new contract alright but player 2 plays either X2, Y2 or S2 okay now if that means in short only player 1 is playing this is signing this contract alright if only player 1 is signing this contract. So, what is happening here effectively what this is saying is that so if only player 1 is going to sign the hat walla contract then he is going to play Y1 player 1 is going to play Y1 then player 2 can either play X2, Y2 or just sign the contract and in which case he will if in or sign the earlier contract which is S2 okay. So, that then gives you this alright. So, the default option in case you do not sign this contract is the same as before that you would play Y1 but now that has to be combined with the earlier this thing also with the earlier contract also given that the other player could still sign the earlier contract. So, you get this here is the same as this here and this here is the same as this here okay. So, now you have another game in which there are now there is not just 1 but 2 contracts the players now go there is this option for them that has another option that has come up and a question for them is which one do they sign alright. So, the contract signing game now is so you have this game with these contracts what are the equilibrium of this game. So, you will see actually the earlier contract remains an equilibrium alright but some more equilibrium actually this turns out to be also an equilibrium and it turns out there is one randomized equilibrium also means there is a mixed strategy equilibrium in which with 50% chance they will these guys would flip choose contract the the the earlier contract and 50% chance they choose the Hatwalla contract okay. So, there are actually 3 equilibria here first is that both sign the earlier contract second is that both sign the new contract and then there is a third equilibrium in which both in which this in which players do this right. Now, you can see this what the communication and contracts are actually doing what they what are they doing they are taking the actions that exist okay and enforcing a probability distribution on those set of actions for the players that are signing their contract right. The ones who do not sign the contract are choosing actions from the earlier original set of actions right but the ones who sign the contract they are being bound by a certain probability distribution right. So, if in the Nash equilibrium we remember we said that players are randomizing independently and the reason they were randomizing independently was because they had no choice they had no option to communicate they had no facility to communicate with each other. Now players can implement with communication they can implement strategies that sound like this you know here if both players sign the contract they will toss a coin and choose this combination or this combination you can see you are getting you cannot do this if you are randomizing independently this is requiring a coordination between the two players on a certain subset of their actions right. So, this is it is basically saying 50 percent chance I am going to choose this one here this 6 0 thing here or 50 percent chance I am going to choose this and with no probability I am going to choose any of these for this you need communication alright. So, how do we model a contract then one way of modeling a contract is to basically say well I do not worry about the words that are exchanged and all that or what is written in the contract and so on I only worry about what actions is it choosing with what probability okay. So, in the Markov decision processes literature this is what these are this is what is called if you if you know that you know you take an infinite horizon Markov decision process you can although so many complicated things happen you take melt you know this action that action etcetera so many different time. So, multiple trajectories are actually possible but at the end of the day the final cost is captured in terms of what in terms of what is called the occupation measure right it tells you with what probability are you going to take what action that is it. This is effectively an occupation measure type of type of construction for problems like this. So, any contract enforces a probability distribution on the set of actions of players that sign that contract okay. So, let us suppose x 1 till x n are the set of actions of the players okay and let us say I suppose I write x sub for any for a subset s of the set of players you can write x sub s as just the product of x i okay and we will write x minus i as simply product of j not equal to i x j okay. So, a contract signed by a subset of players s enforces probability distribution on x sub s all right. Now, we will use this notation delta of x sub s is the set of all probability distributions on x on x s okay. So, a contract basically that is once it is signed by the set s of players it enforces a probability distribution on x sub s all right. Now, but we do not know who is going to sign the contract right. So, the contract must specify what my a probability distribution for every subset of players that could potentially sign right. So, a contract therefore a contract is given by tau let us say tau equal to tau sub s where s ranges over all subsets of players where tau s is belongs to this. So, tau s is a probability distribution on x s. Now, any such probability distribution this here is what is called a correlated strategy okay any mu in this is called a correlated strategy all right. In particular you can take mu to be independent across the players then you have the kind of distributions you are getting with the Nash equilibrium in the Nash equilibrium okay. So, this is what is called a correlated strategy. So, in short a contract then gives you a correlated strategy for every subset of players that could potentially sign the contract okay all right. So, now if so let us suppose so now and I will just use x to denote this product x okay. So, this is the same as x sub s okay. Now, if all n players sign a contract contract which enforces a correlated strategy mu strategy mu then the payoff then the utility of the ith player would be what what is the utility of the ith player under this contract. So, the contract says so this is signed by all n players okay mu is signed by all n players. So, what is the utility that player I will get from this ui of mu would be yeah it is just the expectation right it is. So, it will be ui of x mu of x in right. So, every profile is chosen with a probability profile x is chosen with a probability mu of x that is the contract okay. So, this is the probability this is what this is the expected payoff that can be achieved under a correlated strategy mu which is which corresponds to a contract signed by all players all right. So, now let us let us write so remember our goal was to come up with so we as I said our solution concept is not a Nash equilibrium anymore our solution is what are the set of payoffs that can be achieved under all possible contracts that can be signed right what in contract signing games like this what kind of payoffs can be achieved. So, what we want to do is now characterize that. So, let us write u of mu as just this tuple u1 i this okay ui of mu right. So, if you so the set of payoffs is this that so what is this this is the set of payoffs payoffs by contracts signed by all players okay. This is the set of payoffs achievable by contracts signed by all players okay. If everyone is going to sign the contract then this is the set of payoffs that can be achieved. But then everyone need not sign the contract right. So, you could have a situation where some subset of players sign the contract all right. So, then considering that then what is the set of payoffs that can be achieved when players have an option of several contracts and some subset of them could sign each of them and so on okay. So, that is basically the question. So, the set of if you want to characterize the solution set of this game which is what is the set of payoffs that can be achieved when any an arbitrary set of contracts is available to the players all right and they have the option to sign or not sign what is the what is it that can be achieved. So, let us so it turns out this can actually be characterized rather cleanly. So, what you want to do is characterize what is the set of payoffs that can be actually achieved right. So, define this V i as follows this is the minimum of so imagine the following. So, suppose player the other players except for player i sign some contract tau minus i all right and player i does not okay. And what is the what is the min max value that he can get. So, what is the the worst that he can get in that setting. So, that is so here that is so you can write this in two different ways. So, assuming so here one way is to write it as a min max the other is to write it as a max min. So, here this is the payoff that player i will get when others sign tau minus i as a contract and he signs he plays x i okay. So, he is doing max over that of x i and outside there is a min over our time tau minus i this by the min max theorem is can you can also write it as in the opposite order you can write this as max over tau i in this. So, this is just a mixed strategy now for player i because he is only he is signing this signing this thing. So, it is just a mixed strategy of min of and these here now are the worst case actions that player that the other players could take okay. So, this is essentially his kind of security level for the player. So, it is the it is the it is the it is the best case payoff that he can get in the worst case when others sign you know the most adversarial possible contract for him the contract that is most damaging to him all right. So, this is it this this is his you can say a kind of security level. Now, if a contract offers him anything worse than this security level player i will never sign this contract because he knows that regardless of what the others sign okay i can at least get this by not signing anything best just playing a mixed strategy over my my original set of pure strategies right. So, what this suggests is that anything lower than vi cannot possibly be in the set of outcomes achievable with contracts for any player. So, it any if you in so our solution set should not have any value lower than should not have value for player i which is lower than vi okay. So, what this means is that any correlated strategy strategy mu in this must satisfy ui of mu greater than equal to vi for all i for all i and n okay for it for it to be enough for it to be an outcome outcome or solution. So, any correlated strategy mu must satisfy you must satisfy this can someone tell me what is this constraint like vi is a security level but what is this effectively saying similar to the principle essentially this is a participation constraint it is saying that i must get a reserve utility for me to entertain any of your contracts and that reserve utility is the worst that i would get if all of you all the other guys sign their contract but i do not right sorry. So, the min max and max min are there. So, are the same here. So, you can do so if so you can write this is both min max as well as max min. So, this is like if i play my pure strategy okay and the others do whatever they want you know some correlation some correlated thing between themselves and that is equivalent to others doing whatever they want in pure strategies together jointly and i playing some mixed strategy against that so min max this it is it is equivalent not necessarily. So, there will be points mu out that do not satisfy this right knows delta x is the set of all distributions. So, so for it for a for me to get a solution for me to call it a solution it must satisfy this right okay. So, so my solution set so the set of so what are the utilities that can be achieved it so the set of utilities that can be achieved can be achieved is a subset has to be a subset of of this u of mu such that u of u i of mu greater than equal to v i for all i mu is in delta. The main result is in fact this is the subset this is in fact a set it is in fact exactly equal to this. So, in short what you can show is that if so what you have to only argue is that if you take a point in this set it can be achieved through some contract alright and what you have to do is basically just look at the other come up with a contract in which you bind the other players to something like this. So, take the worst case from here the min max achieving tau minus i bind the other guys to that and show that that is what can be so there is some contract which achieved this. So, in fact so this what I said is a subset so I will erase that is equal to this. It is a maybe 10 minutes more or 5 minutes more of the proof but the proof is not important point is that you see not everything can just because you have allowed arbitrary communication between players and complete and binding contracts and all that does not mean players can do whatever they want right at the end of the day whether to sign a contract or not is also a rational decision in all of this. So, that is why you know theory of games is what it is it is not that it is not as straightforward as you think it is. So, this is then your set of so effectively this is the Nash equilibrium equivalent when you allow for arbitrary communication and binding contracts between the players and now the question is what contract would they want to sign. So, what we will do next time is now take away the binding communication binding contract business and just allow communication and then let us you will see that that is that is a little trickier because then just because someone has signed a contract does not mean anything does not mean that he will in fact play what is there in it and what not. So, then you will have to you will realize that it is that is a little more involved.