 Welcome back to this NPTEL course on game theory. In the previous session, we introduced this cooperative games, TU games in particular. In TU games, towards the end of the session, we introduced the idea of imputation. So, let us recall the idea of imputation. An imputation is an allocation of payoffs to the individuals in such a way that the allocation to every player should be at least bigger than v of i, his work if he is individual. So, that is known as individual rationality. So, let me write x is an allocation. So, the first thing is that this xi should be greater than equals to v of i. This is individual rational. The second condition is that the sum of all the allocations should be the total worth. This is collectively rational. So, these two if satisfied then you call it as x satisfying above is imputation. So, once we define this imputation, another notation here, an imputation x is said to dominate an imputation y if there exists a correlation c such that summation i xi i in c is less than equals to vc and xi is bigger than yi for all i in c. So, for some correlation xi is better to yi and satisfying this summation of xi should be always less than equals to vc. Now, there are few points I would like to mention here is that an imputation need not dominate other imputation. An imputation need not dominate another. So, for example, in the majority game the dividing the dollar game in the majority two of them choose same thing. If you look at that game then x is equals to 150, 150 is 0 is one imputation y is another imputation 150, 150 you can see that x does not dominate y. So, that is one point. The other point is that the relation of domination is not transitive. If x dominates y, y dominates z, x need not dominate z. So, the example for this is that in the same game let us say take 0, 180, 120 this dominates imputation 150, 150, 0 and then 150, 150, 0 dominates 90, 0 to 10. So, in fact 90, 0 to 10 dominates 0, 180, 120. So, that means you have a cycle now here cycle with domination thing. So, in fact here your task is to find which correlation is giving this one. So, it is not very difficult to see this one. In fact, what you can say that it is possible that every imputation is dominated by some other imputation. We can actually construct an example where every imputation is dominated by some other imputation I will leave it for you as an exercise to construct such an example. So, now that we have recall the definition of imputation and all now we will go to introduce what is called the core. So, we start with a TU game in characteristic form nv, n is the set of players and v is the worth of correlations. The core, so what we would like to call is core is basically is the set of all imputations coalitionally rational. What it means is the following thing. So, let me write it core nv is set of all x in Rn such that first thing is sigma xi is vn, this is the collectively rational then we need to instead of writing the xi grid then why what I will put it summation xi i in c is always bigger than vc. This is true for every subset c of n. So, this condition is known as a coalitionally rational. Suppose if c is the coalition, the total worth they are getting is summation xi i in c. This should be bigger than or equals to v of c. Then this is true for every c then you call this as a coalitionally rational. Recall if c is singletons this is nothing but individually rational. So, thus the core is nothing but all the imputations which are coalitionally rational. So, now let us start making some consequences of this definition. So, let us start looking at it. So, first I would like to put a coalition c can improve on an allocation x that is x1, x2, xn in Rn if v of c is bigger than summation xi i in c. Suppose v of c is bigger than summation xi then this coalition c can improve this x. So, the total worth is bigger than this individually this thing. So, that is not this implies c can improve on x, c can improve on x if there exists some allocation y such that y is feasible for c and the players in c get strictly higher payoff. So, c can improve on x that means there must be some allocation y, y is feasible for the c and the players in c will get higher payoff in y than x that is what I mean by when you say that c can improve. So, we also say that the coalition c blocks x. So, the same condition we can also say that because c is better for the the coalition c is can improve on this allocation x, c can improve on x means c is actually blocking this x. So, in fact what we can say that an allocation x is said to be in core if and only if x is feasible for n and no coalition can improve can improve upon it. So, this is all this is straight forward from the definition if there is something in if some coalition is improving it then that is blocking it. So, such a thing is not possible for a core. So, therefore what we the core is essentially to recall it core means x is in core means summation x i i in n this should be v n and summation x i i in c this should be bigger than v c this is true for every subset c n. So, now we start looking at some examples. Before going for the example first I would like to say that suppose if x is not in core that means there is some coalition c such that the players in c would all strictly do better in some other allocation. So, that is what it says. So, now let us look at the examples. Let us look at the divide the dollar game. So, here the core is given by set of all x 1, x 2, x 3 in R 3 such that x 1 plus x 2 plus x 3 this should be 300 then x 1 greater than equals to 0, x 2 greater than equals to 0, x 3 greater than equals to 0. So, this is going to be the core for the versions 1 all three should agree. So, the core is going to be exactly this Butler set. In fact, the core turns out to be infinite here. So, the core is treating all the three players symmetrically that is also another important thing. In version 2 the core is going to be set of all x 1, x 2, x 3 in R 3 such that x 1 plus x 2 is 300 and then x 1 greater than equals to 0, x 2 greater than equals to 0 and then x 3 is equals to 0 because the first two players are or whatever they decide that is going to be the solution mechanism. So, therefore, x 1 plus x 2 they share the entire 300 and the third player gets 0 that is exactly the core for the second version. In the version 3 in fact core I will give it as a exercise for people to look at it is going to be exactly 300, 0, 0. So, we can try the other versions of the divide and call this I will leave it as an exercise. There is a single 10 core. Now, another example. So, we will consider n to be 1, 2, 3 there are three players then v 1 is equals to v 2 equals to v 3, 0 then v 1 2 is 0.25, v 1 3 is 0.5, v 2 3 is 0.75, v 1 2 3 is 1. So, this is a game. Let us recall core means what x is in core nv if and only first is x 1, x 2, x 3 they should be greater than equals to 0 then x 1 plus x 2 should be bigger than 0.25, x 1 plus x 3 should be bigger than 0.5, x 2 plus x 3 should be bigger than 0.75 then x 1 plus x 2 plus x 3 is equals to 1. So, this is the individually rational, these are the coalescently rational with 2, with 3 when 3 is there then this should be 1. So, if we calculate x 1 plus x 2 greater than equals to 0.25, x 1 plus x 3 all these things if you put it. So, we can actually say is that x 1 has to be less than or equals to 0.25, x 2 has to be less than equals to 0.5, x 3 less than equals to 0.75. So, that comes from x 1 plus x 2 plus x 3 is 1, therefore x 1 plus x 2 is nothing but 1 minus x 3. So, 0.25 is said and minus x 3 other side therefore x 3 has to be less than or equals to 0.75, similarly these inequalities. So, and then we know x 1 less than equals to this x 2 less than 1 and then now you can completely picturize the core. In fact, we can write down the core as a following thing it actually something like this. So, this is going to be the x 1 is equals to 0.25, this is nothing but x 2 is equals to 0.5, this is nothing but x 3 is equals to 0.75, now this point is 0.25, 0.5, 0.75 and this is nothing but 0.25, 0.5. So, in fact this is going to be the core. In fact, we can write down all the points it will be this thing because if we write down the equations and all see that this particular shape is this thing. So, we can actually calculate the core here. So, now let us look at another example. So, this is from the classical book of von Neumann and Morgenstern. So, here player 1 has a house which she values say rupees 1 million and wishes to sell. There are two potential buyers, player 2 and 3. So, we have a valuation is rupees 2 million and also have in each. So, they value this at rupees 2 million and also they have 2 million each. Suppose player 1 sells to player 2 at p where 1 less than equals to p less than equals to 2. The utility of player 1 is p, utility of player 2 is going to be 2 minus p plus 2. This is going to be 4 minus p is his utility and then the player 3's utility is because he still has only 2 million. So, therefore, his utility is just 2. So, a similar thing for if player 1 sells to player 3. So, this becomes a player 1 sells to player 3 again the same kind of thing. So, in that once we what we have got is the following thing v of 1 is 1, v of 2 is 2, v of 3 is 2. So, individually they have the houses player 1 is valuing at 1 million and player 2 and player 3 have 2 million each with them. So, that is this thing and v of 1, if player 1 sells to this thing or if player 1 sells to 3 or player 2 and 3 together then the valuation is going to be 4 and v 1, 2, 3 is going to be 6. So, this is going to be the characteristic form of this game and so X is in core if and only if the following thing happens X1 greater than equals to 1, X2 greater than equals to 2, X3 greater than equals to 2 then X1 plus X2 should be greater than equals to 4, X2 plus X3 greater than equals to 4, X1 plus X3 greater than equals to 4 then X1 plus X2 plus X3 is 6. In fact, you can say that the only thing which satisfies all these things is 2, 2, 2 this is in the core core is a singleton here. In fact, there is another very famous example very commonly mentioned example is called glow market. So, this is a simple case there are 5 suppliers of glows there are 5 suppliers of glows. So, of this first 2 can each supply one left glow and the other 3 can supply one right glow each. So, the one supplying NL is let us assume the first 2 players 1 and 2 the right glow is supplied by let us say 3, 4, 5. The worth of each correlation is the number of matched pairs that it can assemble. So, what it means is that suppose if C is equals to 1 and 3 the worth of this C is going to be 1 because they can combine the left and right glow and then therefore they matched one pair. So, that is this thing. Now, if C is equals to 3, 4 then V of C is 0. So, in fact in general we can say that V of C is going to be minimum of mod C intersection NL comma mod C intersection NR the number of intersections of C with NL and intersections with NR and then take the minimum of those 2 that is going to be VC this is 2 for any subset of N. So, the core and actually you can verify that. So, this is a example in fact we can look at multiple situations here. Suppose if I take NL to be 1, 2, 3, NR to be 4, 5. So, again the core is going to be again single term. So, for example if I take NL to be 1, 2, NR to be 3, 4 if we take it only 4 players if you take it and the cardinality of this thing let us say in fact the core is going to be half, half, half, half again single term. In fact, you can actually start working with variety of situations here itself. So, let us take N is equals to take this much and of this is many players supply one left glove each others supply one right glove each. The core is basically xi is equals to 1 if i is in NL, 0 if i is in NR. So, we can verify that this is again has a unique this thing. So, there are many such examples we can construct the core is a very important concept in this set of games. We will conclude this session with this and we will continue in the next session. Thank you.