 Welcome back to this NPTEL course on game theory. In the previous session we introduced the core and discussed some examples. So now we continue the study of this core. The most important thing we would like to say that the core if the allocation is not in core that means there is some correlation which can be improved. So this is a very important fact and if an allocation belongs to the core that means for each player unilateral deviation will not make player strictly better off. If once something is in core no player would unilaterally deviate because he cannot change it without decreasing someone else's payoff. So thus the core is a very interesting concept. Now we would like to say when the core is non-empty. So core need not be non-empty all the time. So there is some characterization of the core non-empty. So let us we will start describing this characterization of games with non-empty core. So this basically it is due to Shapley and Bondareva independently. So we consider the following linear programming. So let us say we are given this transferable utility game nv and look at this min x1 plus x2 plus so on so xn subject to summation xi inc this is bigger than or equals to vc this is true for every subset c in n and then of course x1, x2, xn is in rn. So basically we are trying to minimize x1 plus x2 plus xn such that summation xi is bigger than vc where the summation is taken over all inc and c is a subset of n. So now this is basically set of linear inequalities and then you are minimizing a linear function. This is a linear. Once you take this thing this is basically a kind of an allocation once you take this one. So what is really happening is that which is the this LP determines the minimum amount of transferable utility which is necessary for an allocation so that no correlation can improve upon. So this LP definitely has a solution if it is feasible if this linear program is feasible then there will be a solution and this is because all the inequalities are greater than type and also there is a structure which makes it feasible. For example if you take this one summation xi has to be vc all of them are this thing and you are minimizing it in fact you can say that this is a x1 plus x2 plus xn each x1 is bigger than or equals to this thing. So therefore you can easily say that there is a lower bound certainly and then even mathematically one can see that there is a solution to this problem. So in fact one can verify that this is feasible also from the linear programming argument. So let us take let x1 star x2 star xn star let us say this is optimal solution of let me call this as LP here this is the linear program of this so what would like to say is that x1 star plus x2 star plus xn star this is certainly bigger than or equals to vn. So this is the main thing that is why the difference is what you can change. Now there are two possibilities one is this x1 star plus x2 star plus xn star is strictly greater than vn and in this case the core is empty. The second thing is x1 star plus x2 star plus xn star is equals to vn then in fact x1 star x2 star xn star this in fact belongs to the core of nv. So in fact as I said let us go back to this one why does this linear program have a solution. So even if we simply look at it you are looking at the linear equations which are above something and then you are looking at the sum of entire thing to be minimum. So all of them for example v1 plus v2 plus vn is a lower bound to that one and then you are looking at infimum among them so they are certainly lower there is a lower bound for that and then you are looking at the infimum so this x1 plus x2 plus xn having that one. So therefore one can easily see that this program even just purely looking at from an optimization point of view you can say that there is a solution. Now once you have the solution so let us call the solution x1 star x2 star xn star now the sum of that by this inequality constraint we know that x1 star plus x2 star plus xn star has to be bigger than vn. Now if there is some solution where this x1 star plus x2 star plus xn star is equals to vn and satisfying this then that should be the solution because you are looking at the minimum. If every solution of this one if the sum is bigger than vn that means the core has to be empty so that is basically the what these two points are saying. So if x1 star plus x2 star plus xn star is strictly bigger than vn the core is going to be empty if that is equals to vn then core will be missing and in fact not that this can never be less than vn because by the very definition of the linear program this has to be greater than equals to vn. Now in fact in the previous examples whatever we have discussed we can actually try to write down the linear program and see what is really happening so I will not go into those details. Now we would like to ask when the core is non-empty. So let us look at the linear program so our linear program the primal problem let me write it is minimum summation xi i in n subject to summation xi i in c is greater than equals to v of c, c is contained in n and then x is in of course Rn. Now if we write down the dual of this problem it actually becomes the maximum of for each coalescence c we have a number alpha c vc and then this summation is over n for each of this inequality you have a number and then you write down that and then this is subject to summation alpha c is equals to 1 of course c is c contains i and this should be true for every i in n and alpha c greater than or equals to 0 for all c. So here I am using the LP duality results so if you write down the dual for this problem so this for each inequality you have a number alpha c so that is what I am saying here alpha c for each c each inequality that is indexed by c alpha c into vc is basically the total worth vc and n into the summation of alpha c for each i all those c contained in i that should be 1 and alpha c I will not go into this LP results but let us look at this. So now we have this strong duality theory the linear programming we have a strong duality theorem so if primal has an optimal solution then the dual has an optimal solution and the optimal values are same of course I am not going to give the details of this duality theorem so if we apply the duality theorem the primal has a solution then the dual has a solution in fact you can also say that if the dual has a solution then primal also has a solution and in such a case the optimal values are going to be same. So now in this present case we know that the primal problem has a solution so therefore the dual problem will also have a solution and then the two values are going to be equal. So we apply this strong duality theorem that means there exists an x star in Rn and alpha star c for each subset c in n so that we have the following thing let me write down summation xi star i in c is greater than equals to v of c this is there and alpha star c is greater than equals to 0 this is again true for every c contained in n and not only that alpha star c summation of alpha star c is 1 for all c containing i this is true for every i in n so this is going to happen so we are basically writing then these conditions if we write down those conditions these are the things the solutions x star and alpha star they should satisfy this one. Now let us now look at the following thing so summation alpha c is equals to 1 with c contains i for every i contained in n this is the condition that we have this implies c c contains i alpha c vc is less than equals to vn. So this is what we are saying this is the feasibility of the dual problem the feasibility of the dual implies the objective function of the dual is less than equals to vn. So this is exactly the what we are looking at it this particular thing so the this alpha c vc is less than equals to vn. So now of course sorry I think there is a small error here this is see this is subset of n so if this feasibility implies this if I take this as a condition then what we can easily say here now is that the optimal value is going to be vn because we know that the optimal value has to be bigger than or equals to vn and here we are saying this is less than equals to vn therefore both should be same vn is there so under this condition this automatically implies in fact this is going to be a necessary and sufficient condition so whatever this condition is basically characterizes just the non-emptiness of the core. So this is basically known as a necessary and sufficient this is known as a balanced this condition is known as a balanced condition. So let us now define the games so a tu game nv is called balance if and only if c contains i alpha ci is equals to 1 implies summation c contained in n alpha c vc is less than or equals to vn this condition is balanced condition that we introduced if a game satisfies this condition then you call that as a balanced game. So what we have proved is that every balanced game has a non-empty core. So when this condition is satisfied we call this game as a balanced game and what we have proved says that a balanced game has non-empty core. So now we switch to another important solution concept for cooperative games which is called Shapley value so now we will switch our attention to Shapley's. So now let us get we are interested in now Shapley value. So before I start about this Shapley value let me just recall that in this cooperative games the core can be empty or non-empty even if it is non-empty the core can be infinite set. So therefore exactly choosing a correct solution is a non-trivial fact. So the Shapley value is another solution concept which is we can define irrespective of what kind of cooperative whether the game is balanced or not so it always defines a unique value. So we will now spend time on discussing this Shapley value. So let us look at it. So let us start with the cooperative game and let us take we are interested in developing a solution concept PV which is basically the allocation for each player the Shapley develops a solution concept which basically satisfies certain axioms. The whole idea here is that these the solution concept satisfying these axioms is uniquely given by the Shapley's value. So we will now discuss these axioms. So Shapley's axioms we will start discussing. Let us take a permutation we start with the game NY let us say pi is a permutation on N then consider N pi V is a game such that the following thing pi V of pi i i in C is nothing but VC. Basically what we are doing is that we are permutating the permuting the players instead of giving 1 to N let us say we give them index pi i pi 2 pi N then their worth of that correlation is basically given by the original ranking C. So pi i i belongs to C is the let us say permuted names of those players then the original names is given by C itself therefore the for them the worth of this particular permuted correlation is exactly VC. So now this is so that means the role of the player i is essentially same as the role of the player pi i. So this is a new game so what Shapley before I go further let us introduce a simple example so that this concept will be illustrated better let us take N is equals to 1 2 3 3 player game and then let us define pi 1 to be 3 pi 2 to be let us say 1 pi 3 to be 2 now N pi V is going to be the following thing pi V of 1 is nothing but 1 is a new index for the original player 2 so therefore pi V 1 is simply V 2. Similarly pi V 2 the 2 is the new index so that is given to the original player 3 so therefore this is going to be V 3 and then similarly pi V 3 the 3 is the new index that is given to the player 1 so therefore this is V 1 so like that we can look at the other situations for example pi V of 1 2 so 1 is the index of the player 2 2 is the index of player 2 3 therefore this is nothing but V 2 3 so like that we can do this is basically the idea of this this thing so the axioms are basically the following thing symmetry linearity then carrier so these are the axiom let us start explaining these axioms symmetry for any V in R 2 power N minus 1 so take any characteristic form V then any permutation pi on N and any player i in N so what I am saying the pi i of pi V is same as phi i V so what it says is that it does not matter the player what name you are giving it is whatever he gets he will get he will get the same thing in any permutated situation whatever name you give it to him he will always get the same thing this is the symmetry assumption so the second assumption second axiom is linearity so let us consider two games N V and N W and define P let us say P is a number between 0 and 1 now consider N P V plus 1 minus P W new game where P V plus 1 minus P W of C is nothing but P V C plus 1 minus P W C let us say this is defined for every C in N so we are if we are adding two games and the worth is just some of the individual word so what the linearity says is that if a player i in this sum of these two games P V plus 1 minus P W whatever he gets is nothing but the sum of what he gets individually that is P phi i V plus 1 minus P phi i W so note that once again here like in Nash bargaining thing we are defining the solution concepts through certain axioms so we are defining this solution procedure for all classes of games for any game you take it and what then you assign some value and that should satisfy certain axioms so that solution rule should be linear here so this is the of course here we are only taking the convex combinations but we can always look at directly also without convex combination here so let us look at the other assumption the carrier so a correlation D is said to be carrier of this correlation game and V if V C intersection D is same as V C for all C intended N okay so what it is saying is that if V of C intersection D D is a correlation which we want to call it as a carrier if then V of C is same as V of C intersection D so whenever D you intersect with D the worth of that is not going to change V of C intersection D is same as V of C this is true for every C okay so now let us look at it look at the following thing what exactly does it say let us say D is a carrier let us take I not in D then what is V I V I is nothing but V I intersection D because D is carrier and I D is not in I therefore the intersection I is empty so this is nothing but V of empty that means this is 0 that means if a player is not there in this carrier that means he is going to get a 0 value okay his worth is 0 therefore so if D is a carrier all players J in outside D are dummy players okay because their entry into the correlation does not add anything when they join any correlation they are not improving the worth of the correlation so in fact you can say that V of C union I is same as V of C for all I not in D this again follows from the same this thing so we if I is not in D then by adding I to a correlation C then you are not improving the worth of it okay D is the basically the only players who can influence some worth so that is less thing so the carrier axiom is the following thing okay V I V is 0 value for all I not in D if a player is dummy then he should get 0 value so this these are the axioms that sharply introduced then using these axioms he characterizes that his value sharply value as a unique solution concept satisfying all these three axioms will continue the proof of this shapely result in the next lecture and today we will stop with this thank you