 Welcome back to this NPTEL course on game theory. In the previous session we have introduced correlated equilibrium where the cooperation is through some common random device. Now we will look at another cooperative aspect which makes the non-cooperative game theory as a cooperative. So this is actually known as the Nash bargaining problem. So in this bargaining problem for example there are two players let us, we start with two players as consider the prisoner's dilemma. In a prisoner's dilemma when they cooperate they are actually achieving a very bad Nash equilibrium. In the Nash equilibrium they are actually that is a inferior equilibrium. But if there is somehow if they can cooperate themselves that means they are not giving testimony on the others then they know that they will get a better one but how do we achieve that? So in some sense can we really come up with certain contracts or something like that which will ensure that they will get a better pay than the Nash equilibrium in for example prisoner's dilemma. So the Nash introduced this problem called Nash bargaining problem so which we will discuss now. So in the Nash bargaining problem this is a situation where the following thing there are individuals they have the possibility of concluding mutually beneficial agreements. So they know that by cooperating they can get a better, better utilities. So the second point is that there is a conflict of interest about the agreements about which agreement to conclude. So among all the mutually beneficial agreement if they have everyone is okay with something compared to the other then they will go for it. But the most important thing here in the Nash bargaining problem is that there is a conflict of interest. So what is good for one player may not be good for the other player. So that is another thing okay and this agreement no agreement without player approval. So whatever agreement that we are considering the player should agree to this without their agreement nothing will be there. These are the main points of the Nash bargaining. So there are certain, certain necessary information when this thing is there if for example if there is no agreement we must specify what the players. So this is the disagreement payoff okay. This is a disagreement payoff if the players are not coming to agreement how much they will get it. This is one very important to specify in this thing. Another important point to specify here is that possible agreements this is there. So now this situation possible agreement the possible payoffs that how much they are getting that this is another thing to specify in this problem. Now there are several places where such situations arise for example the management labor attrition where the management negotiates with the labor union. So basically labor union have certain utility over what agreement they should go with the management and that is one very nice example where this subject has been applied. In fact there are several other possibilities for example even in duopoly market when the two firms are competing in a market they can actually go for this bargaining. So that both of them can benefit by staying in the market. So that they can avoid the non-cooperation and hence inferior equilibrium. So now let us formally describe this problem. Let me this is basically introduced by Nash. So this consists of pair F and V where F is basically feasible set of allocations. So this is basically F is a subset of R2 and we assume that this is a closed convex. What is V? V is the degree of agreement point, V is nothing but V1, V2, this is in R2. So what it says is that if no agreement player 1 will get V1, player 2 will get V2. So this is basically the, so now we need to there are certain assumption that we have made closed and convex of course there is one more thing is that we need to make sure that V is in F but this is not a serious assumption but V should be a possible allocation vector. So therefore in that sense it is natural to assume this one. Now there is one more thing we need to take the set F intersection set of all x1, x2 in R2 such that x1 greater than equals to V1, x2 greater than equals to V2. So basically because V1, V2 is the disagreement point there must be a allocation where both the players get higher than V1 and V2 there must be, so that means this set must be non-empty. If all the points in F are less than V1 and V2 they would not go certainly for this bargaining. So now let us look at some justifications. So why convex, this for example if there are two agreements, two allocations possible and now players can try to look at a randomization. Now when you look at the randomization the feasible allocation then will be a convex combination of two points in F that should also be there in F. So this is in that sense is a technical assumption to include the randomizations like that. So therefore the convexity is necessary then F is assumed to be close. So this is basically closeness is a natural topological argument for example if we have a sequence of agreement points and then their limit should also be there. So this many a times will become a mathematical necessity. So therefore this thing and as I said this particular set non-emptiness this is necessary because if there is no vector in F which satisfies these two conditions then there is no bargaining at all, no one would agree whatever it is. So therefore V1, V2 is going to be their pay of automatically, so therefore this comes. So these are the reasons why this thing. Now let us look at how these are connected with non-zero sum games, connection with say bi-matrix games. So now let us take a game there are two players S1, S2 are the strategy spaces U1, U2 are their payoffs so let us consider this game. Now let us look at the, let us take F to be the set of allocations under correlated strategies. So that is nothing but U1 mu, U2 mu such that mu is a correlated strategy. So this is basically a, this thing recall U1 mu is nothing but summation S in S, mu of S, U1 S and this is nothing but summation S in S, mu S, U2 S and you can easily see that the F is a convex and compact, in fact close it set because the S1, S2 are finite strategies so therefore PS1 cross S2 is compact so therefore mu is compact so therefore you can say they are all compact, F is a compact set that is in fact what we need is closeness and F is this thing and how can we choose V? So V could be an inash equilibrium for example, there are several possibilities for disagreement vector there are multiple ways to choose V. For example V could be there mean max strategies one possibility is V1 is minimum sigma 2 delta S2 max the mixture strategies of player 1 this is for player 2 U1 sigma 1 sigma 2 this is the V1, V2 will be again the corresponding thing so this is one possibility the mean max this thing is the in the worst case they will be getting this much so V1, V2 could be one disagreement vector or the another possibility is choose Nash payoff so Nash payoff vector is another possibility and then there are many other ways to choose it. The Nash actually solve this problem using certain axioms so let us write down the axioms of Nash we will start describing them. So there are five axioms one is strong efficiency first I will write and then I will start explaining everything individual rationality scale covariance independence of irrelevant alternatives the other one is symmetry. So these are the five axioms that Nash proposed then we need to explain each of them but let us use the following thing. So let us use F, V is a Nash bargaining problem FV is basically nothing but F1, FV, F2, FV this is the solution the Nash bargaining solution the concept that we are going to define. So you given a problem F, V then the solution of this we are denoting by F FE, FE is basically the rule that it is giving the solution F1, FV, F2, FV. Now this particular rule how we were defining FFV that should satisfy certain axioms and now we need to explain this axioms. So what is strong axioms efficiency first let us look at it so we are already given F. So what is a given allocation X is equals to X1, X2 in F this is basically strongly Pareto efficient we say this X to be a strongly Pareto efficient if there exists no Y that is Y1, Y2 in F such that the following holds Y1 is bigger than or equals to X1, Y2 is bigger than or equals to X2 with strict inequality satisfied for at least one inequality. So that means X1, X2 there is no Y which has higher than X1 or higher than X2 and one of them should be strict and other could be equal. So if there is no such Y then we say this allocation X has a strongly Pareto efficient. So we will also say that an allocation X that is X1, X2 in F is weakly Pareto efficient if there exists no Y such that Y1 greater than X1, Y2 greater than X2. So Y1 should not be strictly bigger than X1, Y2 should not be strictly bigger than X2 then you call this X1, X2 as a weakly Pareto efficient. Now let us the strong efficiency axiom the axiom assets that solution of bargaining problem be feasible and strongly efficient. So why do we require this? So the solution should be strongly efficient. If it is not strongly efficient that means what both the players actually have a other option where basically there is something both of them will get higher payoffs. So therefore they would like to go for that instead of this one. So therefore assuming that the solution should be strongly efficient is a quite a natural axiom that is the strong efficient. Now let us look at the individual rationality. So we will now consider the individual rationality. What is says that Fv should be greater than equals to V that means F1 Fv is bigger than equals to V1 F2 Fv should be bigger than equals to V2. This is again natural because if any player is getting less than the disagreement pay off they would not like to consider this solution. So this is an individually rational this thing that means each player would like to get at least V1 if not more. So this is the individual rational axiom then the next scale covariance. So this depends on some relation let us take lambda 1 lambda 2 mu 1 mu 2 with lambda 1 strictly greater than 0 lambda 2 strictly greater than 0. Let us consider few numbers lambda 1 lambda 2 mu 1 mu 2 with lambda 1 lambda 2 strictly greater than 0. Now define G to be lambda 1 x 1 plus mu 1 lambda 2 x 2 plus mu 2 such that x 1 x 2 is in F look at this set. So what is really happening is that the G is nothing but a scaled version of this vector x 1 x 2. So x 1 lambda 1 x 1 lambda 1 is positive and lambda 2 is also positive. Now then I will take another point W this is also scaled lambda 1 V1 plus mu 1 lambda 2 V2 plus mu 2. This is the this thing now consider Gw the new bargaining problem. So what should be the solution of this? So we can easily say is that F of Gw should be same as lambda 1 F1 Fv plus mu 1 lambda 2 F2 Fv plus mu 2. This is a natural to again expect for example all the numbers are scaled by some constant let us say 2 then everyone whatever you get in F in G they must get double to that. So that is exactly what it is saying multiplied by lambda 1 lambda 2 and then this mu 1 is the translation. So this is the scale covariance when all the vectors are scaled and translated by something the solution should also get have the same effect. So this is the scale covariance then the next assumption axiom is independence of irrelevant axiom. So now this axiom is that suppose for any closed convex G such that for any closed convex G such that G is contained in F and the solution corresponding to F is also in G then the solution corresponding to Gv should be same as. So what is this assumption? Let us say that if Fv is your bargaining problem the solution of Fv actually is inside G then if you consider the restricted agreements G with the same disagreement vector V then the solution should be same as that. So this is a known as independence of irrelevant axiom. In other words what you are saying is that if certain allocations from F are removed which are not solutions in such a way that the G still happens to be convex then the solution will also continue to be same. So this is the axiom 4 then the final axiom is symmetry. So this axiom says the following suppose V1 their disagreement payoffs are same and also if set of all x2, x1 such that x1, x2 is in F basically we are taking the symmetric this is same as F that means if x1, x2 is in F then x2, x1 is also in F. So the symmetric of this thing then the solution should also have symmetric F1, Fv should be same as F2. So when V1, V2 the disagreement payoffs are same and the set F is symmetric then the solution should also satisfy that the play whatever player 1 gets same player 2 will also get. So these are the 5 axioms under these 5 axioms Nash proposed the following theorem. So let me state the theorem given when a 2% bargaining problem Fv there exists a unique solution F that satisfies the 5 axioms. In fact the solution satisfies the following thing. So what is this is that Fv belongs to arg max x1, x2 in F, x1 greater than equals to V1, x2 greater than equals to V2 of x1 minus V1 into x2 minus V2. So this is the theorem that Nash proved what this theorem says that if I define the solution rule F of Fv to be a the vector which maximizes this product x1 minus V1 into x2 minus V2 over the feasible allocation satisfying this individual rationality then that solution rule satisfies this 5 axioms and this is the unique solution. We will prove this theorem in the next session and then we stop with this. Thank you.