 Welcome back to NPTEL course on game theory. In the previous session we have introduced Nash bargaining problem and we derived the Nash bargaining rule satisfying certain axioms. Now we will look at one example to illustrate certain aspects. So, consider this v the disagreement vector v to be 0 0 and then I take f to be set of all y 1 y 2 such that 0 less than equals to y 1 less than equals to 30 and 0 less than equals to y 2 less than equals to 30 minus y 1. So, in a way what we are saying is that y 1 plus y 2 has to be less than equals to 30. Now, this is a situation where the two players are going to choose 30 units among themselves. So, how do this split? Now no one wants to get less than 0 that is basically the disagreement vector. Now let us assume that the players have a value the money linearly. So, this essentially means that the players are risk neutral. So, the players have a linear utility for money. If they have 5 rupees that whatever worth for them then 10 is exactly double to that. So, they have a linear utility this means that the players are risk neutral. So, this thing in fact, in this case, the solution you can actually look at from the symmetry or whatever way you can calculate the solution to be 15. It is not hard to see this one the symmetry and y 1 plus y 2 less than equals to 30. So, it is a symmetric f is symmetric. So, therefore in a symmetric thing y 1, y 2 has to be same and then they can go up to 30. So, therefore y 1 and y 2 both will be equal to 15. Now that is the same problem, but now assume player 1 is risk neutral but the player 2 is risk averse. So, here utility is basically proportional to square root of f 4 units of money means it will have a utility 2. So, in this case f becomes y 1, y 2 is at 0 less than equals to y 1 less than equals to 30. The player 1 has a linear utility whereas the player 2 has a square root 30 minus y 1 is the same they are splitting the same 30 units but then the player 2's utility is like root 30 minus y 1. If he gets 30 minus y 1 then his utility is root 30 minus y 1. Now disagreement vector is 0 0. Now in this case we will leave it to you as an exercise the solution is going to be 20 and root 10. So, in a sense this is corresponding to the wealth sharing of 20 and 10. So this is again another interesting example illustrating this Nash bargaining rule. Now we will look at certain other concepts. So, there are two aspects here one is called principle of equal gains what it says that you should do this for me because I am doing this. So, this is known as a principle of equal gains then the second thing is principle of greatest good. So, here the argument is that you should do this because it helps me more than it hurts you. So, these are the two principles and in the first principle principle of equal gains gives what is called egalitarian solution and in the second principle gives you utilitarian solution. So, we will describe these solutions. So, let us take the bargaining problem Fv the egalitarian solution shown Fv basically this is unique point x 1, x 2 in F that is weakly efficient and satisfies the following condition. So, x 1 minus y v 1 is what I am getting it this should be same as x 2 minus v 2 because you are getting x 2 minus v 2 I should also get the same amount that is essentially this thing. So, this is known as the egalitarian solution anything this satisfying this. Now what is the utilitarian situation utilitarian? So basically here Fv this is x 1, x 2 in F satisfies such that this x 1 plus x 2 should be the max over y 1, y 2 in F y 1 plus y 2 you take the maximum of that this is going to be this thing. So, this is going to be the utilitarian solution basically because I am getting the maximum possible not at a cost of yours it is not hurting you. So, therefore you are maximizing y 1 plus y 2. So, both thus in fact one thing is that both violates axiom of scale covariance they violate the axiom of scale covariance. In fact, it is not hard to see it I will leave it for you to verify this one can consider some simple examples. Now we would like to write what is called lambda egalitarian solution basically this is given numbers lambda 1, lambda 2, mu 1, mu 2 with lambda 1 greater than 0, lambda 2 greater than 0. So, let L y is lambda 1, y 1 plus mu 1, lambda 2, y 2 plus mu 2 of course y in R 2 now given F v. So, L f is nothing but L y such that y in F. So, let us consider the egalitarian solution of L f, L v is L x where x is unique, weakly efficient point in the F such that the following thing satisfied lambda 1 x 1 minus v 1 should be same as lambda 2 x 2 minus v 2. So, you find this one this is known as this is called lambda egalitarian solution of F v. So, basically this lambda 1 and lambda 2 are giving you the what they are considering the ratios. Of course, when lambda is equals to 1, 1 this is simple egalitarian solution. So, as we said this they already we said their roads satisfy the scale covariance. Now, let us look at the lambda egalitarian. So, because L f, L v take this one the point is L z where z is equals to z 1, z 2 in F such that lambda 1 z 1 plus lambda 2 z 2 is max y 1 y 2 in F such that lambda 1 y 1 plus lambda 2 y 2. So, basically y 1 you are giving the lambda 1 scale here y 2 you are multiplying by lambda 2 and look at the maximum of them that corresponding one is lambda 1 z 1 lambda 2. So, this is basically the lambda egalitarian solution this z is the lambda egalitarian solution. Now, what is the relationship Nash Bargaining? So, let I will just make a statement here let F v be the essential two person bargaining problem. Suppose this star is in F and x star greater than equals to v then x star is Nash Bargaining solution for F v if and only if there exists strictly positive numbers lambda 1 lambda 2 such that lambda 1 x 1 star minus lambda 1 v 1 is equals to lambda 2 x 2 star minus lambda 2 v 2 this and lambda 1 x 1 star minus lambda 2 x 2 star is same as max y in F lambda 1 y 1 plus lambda 2 y 2. So, in a sense what we are saying is that the Nash Bargaining solution is same as the lambda egalitarian solution as well as lambda egalitarian solution for some lambda 1 lambda 2. So, such a thing exists the proof we will not go into the details we can try working on our own. So, I will leave it for the people who are interested to work on it. So, this gives you some idea about the Nash Bargaining problem with two players. Now, we would like to switch our attention to the multi person bargaining problem. So, let us look at multi person bargaining problem. So, what exactly is the multiple? So, there is basically we take n to be one there are n players. So, there are n is now we are assuming it n is bigger than 2 then let us say F is some subset of R n and V is in R n. So, F v is basically the multi person bargaining problem. From the previous session we can think that we take this pi i in i xi minus vi we can define the Nash product, Nash Bargaining product and we can say that we can think about this solution. But there are reasons why this solution is not good. Before going further let me mention one thing is that in the previous session when we are proving the Nash Bargaining problem and this product has a unique maximizer we said that this is a quasi concave function. But there is another way to see this result is that instead of taking the quasi concave t you take the logarithmic transformation of this one. So, that means log x1 minus v1 into x2 minus v2 that becomes log x1 minus v1 plus log x2 minus v2. So, the logarithmic transformation when you apply it whatever maximizes the horizontal problem should also maximize the log x1 minus v1 plus log x2 minus v2 use the properties of log. Now the solution of log x1 minus v1 plus log x2 minus v2 comes easily because they are now separated in the two variables and then we can actually apply anything convexity and other things and we can get the same result. So, now let us look at come back to this multi person thing. When you look at this Nash Bargaining product for multiple thing the problem is that this thing does not consider all possible coalitions. What I mean to say that if for example when there is two person this thing there are only two ways the players are cooperating themselves and then getting it whatever is now here there are multiple possibilities of corporations. So, for example any subset of n can cooperate together from the rest. So, those aspects are not taken care by this product. So, in fact what we will do is we take a simple example and look for different solution concepts within the same. So, let us see some examples. So, what we look at is it is a very classical example it is known as a divide the dollar we consider multiple solution to the same problem. So, let us take there are three players. So, the players have to divide a total wealth 300 they have to divide this 300 units among themselves. Now, each player can propose a payoff such that no players payoff is negative and the sum of all should not exceed 300. Therefore, the strategy sets for them is S1, S2, S3 all of them is basically X1, X2, X3 in R3 such that X1 plus X2 plus X3 is less than equals to 300 then all of them should be non-negative. So, these are the strategy space. So, everyone specifies an allocation rule X1, X2, X3 and player 2 will also specify his own this thing and then this thing. Now, there are multiple ways you can now look at it. For example, if everyone should choose same allocation, otherwise they get 0. In other words, what we are saying the utility for them is that the player 1 chooses the allocation vector S1, player 2 chooses allocation vector S2, player 3 chooses S3 and this the player 1's utility is going to be XI if all of them are same is 0 if not. So, now, in fact, we can verify that here the solution is 100. So, this is like a 3 percent bar guiding problem and we can see that the solution in this case will be 100, 100, 100. So, which is a reasonable solution in this model. Now, let us look at another version. So, here, so players get 0 unless player 1 and 2 propose same allocation. If player 1 and player 2 proposes the same allocation, then they will get whatever they have proposed, otherwise they get nothing. So, in that sense, Ui S1, S2, S3 is going to be XI if S1 is equals to S2 that is same as X1, X2, X3 and 0, otherwise. So, in fact, what will be the equilibrium in this case? So, player 1 and 2 actually will say that they would like to divide this equally between them. So, therefore, 150, 150, 0 is one solution here. So, the way to do this one is that because S1 and S2 are the main people whose decision is affecting their payoff. So, they can think that they can ignore the third person and think it as a 2 percent bar guiding problem and hence this 150, 150, 0 comes to this thing. So, in fact 100, 100, 100 would also be a reasonable solution, but how do we justify this one? So, that is another issue here. So, this is also one solution. So, now look at another version. So, in fact, it is very difficult to justify this solution here. So, there is another way is that there is a slight variation. Let me call it as version 3. So, let me this thing here Ui S1, S2, S3 is XI if S1 is equals to S2 is equals to X1, X2, X3 or S1 is equals to S3 which is X1, X2, X3, 0 otherwise. So, either player 1 and player 2 proposes the same or player 1 proposes the same as one another player, then they get this one. So, in fact I would like look at reasonable solutions in this setup. So, this is another way you can look at it. Then there is one more version is Ui S1, S2, S3 is XI if Sj is equals to Sk is equals to X1, X2, X3 for some j not equals to k and 0 otherwise. What I am saying is that if the two of them are saying the same thing, then that is enforced, that means the majority is considered and if there is no majority then they get 0. So, again look for solutions possible here. So, there are multiple versions we can consider of the same game but what rules. So, in this way actually this multi the bargaining problem that same idea cannot be done. So, it requires some new thoughts on this thing and now this is the place where we would like to introduce what is called games in characteristic form. So, this is TU games transferable utility, what it says that transferable. So, in a way in the previous examples also what we have seen that a player can actually they have to divide for example the 300 among themselves. So, they can give some money to the other and whatever it is the money can be transferred among this. So, such games are known as transferable utility games. So, as we said in the multi person bargaining it is the solution is very difficult. So, we need to come up with different solutions. The solution depends on that for example as a cooperation suppose if a subset of people are together how much worth it is we have to specify. So, we need to consider the correlations among the smallest. So, in that sense what this game is given by the set of players n and there is a function v what it says that n is basically the set of all players the n players then v is basically is a for any subset let us take a subset s of n then v s is telling is the worth of the correlation s. How much s together we will get it this thing and then we also need to say that v of empty if there is no correlation then they are getting 0 worth. So, these are known as a transferable utility. So, in fact we can also define the non-transferable utility games here is basically this is a game on the set of players n is basically a mapping v is mapping from okay and okay I should put it. So, v of c if c is a correlation c is a subset of n. So, v of c should specify each person how much he is getting that means v of c is a subset of r mod c. So, the correlation we are specifying their value each of them how much they are getting it. So, this is the main difference from the transferable utility. So, we assume this is a non-empty closed convex subset. This is the first step and then the second thing is that set of all x such that x is in vc and xi greater than equals to vi for all i in c is bounded subset of r mod c where vi is defined in the following thing max yi such that y is in v. So, v of i is a single term whatever this thing all those y's in vi. So, look at those yi's and then look at the maximum this should be less than infinity for all i in n. So, a n to u game non-transferable utility game is defined by this one. So, in the next session we will start working with several examples of these games and then we continue our study of the these games. Thank you.