 Welcome back to NPTEL course on game theory. In the previous session we have introduced several axioms which are necessary for understanding the Shapley value. In fact we made a statement regarding the Shapley value. So let us start with the theorem and we will try to prove this. This is due to Shapley. So there is exactly one mapping phi from 2n minus 1 to rn that satisfies the axiom. This the map is given by phi iv is nothing but summation over c contained in n minus i mod c factorial n minus mod c minus 1 factorial by n factorial into the marginal contribution of i. So remember the axioms that we wanted is the following thing. The axioms are symmetry, linearity, carrierism or even null player we can say. With these axioms we need to show that this phi iv is the unique allocation rule satisfying these axioms. So let us proceed for the proof. The proof is divided into two parts. In the part 1 we show that the mapping given in the theorem satisfies the axioms. And then the part 2 we prove the uniqueness. We separate these two parts and then let us start with the part 1. This is first we start with the linearity. So what we need to show is that phi i of pv plus 1 minus pw where v and w are 2 cooperative games and p is a number 2011 then what we need to show is p into phi iv plus 1 minus p into phi iw. So this is what we need to show it. So let us this thing by definition for any correlation pv plus 1 minus pw of this c is nothing but pvc plus 1 minus pwc. Now look at the formula for corresponding to pv plus 1 minus pw. So this is nothing but summation the mod c factorial into mod n minus mod c minus 1 factorial by n factorial into pv plus 1 minus pw of c union i minus pv plus 1 minus pw of c. So here this c is contained in the summation is taken over n minus i. So let us basically we need to write down the values of this. So that is going to be same as summation c contained in n minus i mod c factorial into n minus mod c minus 1 factorial by n factorial. If we write it that will be pvc plus 1 minus p wc okay c is basically c union i here c union i minus pvc plus 1 minus pwc. Now I can separate these two things this term and this term and this term and this term. So if I write it separately that becomes pvc vc union i minus vc plus 1 minus p wc union i minus wc. Now because summation over the finite clemeni number so therefore this simply becomes summation c is contained in n minus i of the mod c factorial into n minus mod c minus 1 factorial by n factorial into p comes outside here vc union i minus vc that is the first term. The second term corresponds to 1 minus p into again c contained in n minus i mod c factorial n minus mod c minus 1 factorial by n factorial into wc union i minus wc. Now if we note down this particular term is nothing but the phi iv this second term is phi i w of course with the multiplicative factors p and 1 minus p. So this for this is nothing but p phi iv plus 1 minus p phi i w. So this shows that the mapping phi i is linear. Once we know that this is linear so let us look at the next thing the carrier axioms. Suppose d is a carrier of the game nv. Now therefore vc is nothing but vc intersection d for every c contained in n. Now we also have that v i is 0 for all i outside d and v of d is nothing but v of n. So these are all there. Now if we apply this into the Shapley value this thing in fact it is straight forward to verify phi iv is 0 for all i in n minus d that is outside d. It is directly we can use take i not in n minus d and then substitute there and then we are going to get this directly. Now of course then what we need to show now finally is that for carrier d we need to show i belongs to d phi iv is nothing but vd. Again this comes from the fact that if we sum all the phi i is this phi iv over i in n this is nothing but vn which is same as vd because we have vn and vd are same that is this thing. This actually just substitutions and we can because this also comes from the fact that this is there. So we can actually complete this thing. Of course we have missed to see one more thing is that the symmetry which we should have done first. So symmetry is basically if you really look at it in any correlation phi iv if you take it it depends only on number of players in a correlation c and whether c contains i or not it does not matter really. So therefore the relabeling has no effect on the value therefore the symmetric holds. If we really look at it whether i belongs to c or not that is all is metering because by the formula it depends on vc union i minus vc and you are taking the relations and then it depends only on the number of this thing. So when I relabel the same set of things keep coming so therefore the symmetry automatically holds. So this proves the fact that the three axioms are satisfied by the Shapley value. Now let us go for the proof of 2. So what we will do here is the following thing. The proof really depends on, let us look at the nz be a cooperative game that essence what 0 to every correlation c. So if I take that one that means we are really considering a 0 game in a sense every correlation has the 0 value then what we get here is phi i z is going to be 0. This is by the very definition of the carrier axiom. The carrier axiom tells us that if it is a 0 game then phi i z is always 0 for every i that is one point. Then the other point is that the linearity if you take any two games phi i pv plus 1 minus pw we know that the linearity says that p into phi i v plus 1 minus pw. So that means the map phi i is a linear if for example if a phi i if a phi i is a map satisfying the axioms the linearity carrier and then the efficiency the symmetry then what we have is that this is a linear map. Now therefore the mapping satisfying the axioms is linear. So now once we know that the mapping rule allocation scheme is a linear function over the space of all cooperative games then the way to see that this is a unique the Shapley value is the only map satisfying those three axiom what we can really do is that how the this allocation depends on a basis. Now let us look at the this ln let us say this denotes all subsets of n in fact the cardinality of this ln is 2 power n minus 1. Therefore r mod ln this is the is a 2 power n minus 1 dimensional space basically vector space. So this space contains all the cooperative games. So now let us consider the following simple game let us say d is a subset of n and then define n w d as following thing w d c is 1 if d is contained in c otherwise it is 0. So now how many such games are there for each subset of n we have one game like this and therefore there are 2 power n minus 1 thing. So therefore there are 2 power n minus 1 such games we now show that this forms a basis if we see that this forms a basis then we are done. So how we can show this so because these are all 2 power n minus 1 sets if we can show that they are linearly independent that means then we are done. So let us show that the set of all these 2 power n minus 1 games is linearly independent. So what it means is that if these are all linearly independent how do we what we need to show. So let us take d 1 is one correlation and d 2 is another correlation let us take it and consider the games w d 1 n w d 1 n w d 2 say d 1 is not equals to d 2. So I am just showing the 2 such correlations I take and 2 such games and we need to show that this a linear combination of these games so that is a p w d 1 plus let us say 1 minus p w d 2 is the 0 game then more than p here let us put alpha and beta then alpha is equals to beta is equals to 0 this is what we need to show. So in fact it is not really hard to verify this one because alpha w d 1 will become let us say for any c alpha w d 1 c plus beta w d 2 c is nothing but alpha into so let us it depends on multiple things so let us go back once again to define if d is contained c it is 1. So this is basically 1 let me write it the best way is to put it this is 1 if d 1 contain in c similarly d 2 c is 1 if d 2 is contained in c and 0 otherwise. So therefore whenever d 1 is contained in c and because d 2 and d 1 are not same there are certainly a subset see choose basically c such that d 1 is contained in c but d 2 is not contained in c then this particular value will be simply alpha into 1 plus beta into 0 that is alpha. Now we have assumed that if this is 0 that for any correlation this has to be 0 therefore this has to be 0 so now only thing that I need to show that such a c can be obtained but that is automatically possible because d 1 and d 2 are at different sets so we can always choose some c which is contained in d 1 but not contained in which we can see we can choose c containing d 1 but does not contain d 2 such a c we can always choose. So therefore we can say that this alpha has to be 0 if this is a 0 game similarly beta can be made 0 the same argument I will choose now a correlation c which contains d 2 but not d 1 so therefore beta has to be 0 therefore alpha w d 1 plus beta w d 2 is 0 implies alpha is equals to beta is equals to 0 that means this is a linearly independent set. So in fact I need to say something more before I say that this is a linearly independent I have proved this only for 2 sets but I can do it with any sets any subsets I can take it any number of subsets d 1, d 2, d 3, d k which are all different and then if their linear combination is 0 then I can show that the corresponding coefficients have to be 0. So therefore this is linearly independent now note that how many such games are there. So because for each subset d I can define one game and there are total that 2 power n minus 1 games therefore there are 2 power n minus 1 such games and all of them are independent the set is a independent set. Now this is a 2 power n minus 1 dimensional linear space therefore this is a basis now from a linear algebra we can easily see that any linear function on a vector space if it fixes the values on its basis then such a linear transformation is unique. So therefore to prove the uniqueness it is enough to show that the mapping fixes its values on these simple games. So that is all we need to show let us start looking at it. So choose d a subset of n we have this game w d so phi i of this game what will be this one. Now remember this is where we use a carrier axiom because d is a carrier here this we can easily verify that this value is going to be 1 by mod d. Now we can verify from the formula in theorem the value coincides with this. So basically what we need to show is that going back to this one we have this formula this formula just use instead of v use the simple game w d and show that phi i w d is nothing but 1 by mod d. Once you show that this becomes a 1 it is 1 by mod d that means this whatever this map whatever is defined this is fixing the values on the Shapley value on the simple games which is the basis therefore any linear transformation which fixes the value its values on the basis has to be unique and therefore this proves that this formula is the only map satisfying this theorem this three axioms. So that in that way the value is the unique function satisfying these things. So this is known as the Shapley value in fact it has multiple ways you can do it. Let me write it here phi i v is defined as summation c contained in n minus i mod c factorial into n minus mod c minus 1 factorial by n factorial v c union i minus v c. So basically this is a average of a marginal contribution that a player is contributing to a correlation c and any correlation c outside this n minus i can be chosen in these many ways mod c factorial into n minus mod c minus 1 factorial by n. This is the number of ways that you can choose an equation c not containing i therefore the player's marginal contribution for a correlation c is given by this and now it is weighted some expectation is nothing but the Shapley value. This is a useful interpretation which helps sometimes. So the another very important thing that we can say here is that there are under certain for certain class of games Shapley value belongs to core. So what is such games these are known as a convex games. So the convex game are satisfying the following condition. So v of d union i marginal contribution a player gives the marginal contribution of a player i is always bigger than marginal contribution of the same player for another correlation whenever c is contained in d is contained in n and of course i not in d. So that means what if you take any correlation c and another correlation d which is strictly bigger than c then the marginal contribution that the player i who is not in d is contributing to c is less than his marginal contribution to d that means for larger groups his contribution marginal contribution is higher. So once you define this convex games what we can show is that for convex games Shapley value belongs to. So we will not prove this one but this is a very important result and in the next session we will go to another important concept in cooperative games which is called nucleus we will stop with this today. Thank you.