 Welcome back to this NPTEL course on game theory. In the previous session, we introduced a multipartisan bargaining problem and some difficulties associated with it and then we went to define what is called the transferable utility games and non-transferable utility games. But in the sequel, we will concentrate only on PU games, the cooperative games with characteristic form are transferable utility games. So we will not consider the N2U games. Now we will see some examples of this TU games. So let us look at the dollar game, divide the dollar game because that is where we introduce this multipartisan bargaining game and then multiple solutions possible. So let us look at this thing. So if there are 3 players, all of them together worth is 300, they have to share this 300 rupees and we want, if only 2 people are sharing, we want them to get 0. Then V of singleton 1, V of singleton 2, V of singleton 3, all of them also get 0. So they have to share 300 but no less, this is the version 1. So in fact this is for version 1. The version 2, we said player 1 and 2 are the ones who is making the decision whatever they choose, therefore V of 1, 2, 3 is 300, this is same as V of 1, 2 and any other combinations get, all the other combinations get 0. This is this. So likewise we can actually look at the other 2 versions also. So this gives you the characteristic form representation for the game. So now basically for each such characteristic form, what is the corresponding solution and we will try to build that solution concept in this course. Now let us look at another, this thing. So there is a, let us say another example is voting game. So there is some 4 parties, 1, 2, 3, 4. So the party 1 let us say they have 45 members in the, let us say in the parliament and party 2 has let us say 25, party 3 has 15 members, party 4 has 12 members. So let us say to pass any bill, 51 votes are required. So this situation we can model it as a TU game with 4 players. So of course no single party has enough this thing. So no single party has 51 votes. So therefore single party gets 0 this thing. Now V of 1, 2, 1, 2 together is more than 51. In fact any 2 if you take it they are going to be more than 51. So therefore all of them get, let us say full this thing. So let me put, in fact when I said any 2 is basically 2 and 3 together would get, if 1 joins with any of them then only 51, more than 51 votes are there. So if 1 is not joining with any of them 2 and 3 together we will get 0 and similarly 3, 4 are 4, 2 together but 1 if joins any one of them then they get full. So V 1, 4 and V 1, 3, 4 I will remove the brace symbol here, 1, 3, 4, 1, 2, 4, 2, 3, 4, V 1, 2, 3, 4 all of them has 100 and V 2, 3, V 3, 4, V 2, 4 again get 0. So if the number of votes by the parties is more than 51 then they are getting full 100 and otherwise that correlation gets 0. So this is another model of this voting game. In fact there are several examples that we can come up with this. So let us look at some representations for U games. So we particularly will look at min, max representation. So basically we are considering a strategic form game with transferable utility. Let us say consider a game G where the set of players are N, their strategies sets are S1, S2, SN and their utility functions are UI. Let us consider this one. This is a strategic form game like the games that we have discussed in the non-cooperative games. In fact in much of our analysis we only considered two player games but we can consider a same situations with multiple players, several of the existential results and everything can be proved. So we are considering such strategic form games. And then we are assuming that this is a transferable utility. And let us say C is a coalition, any coalition let us take and then N minus C is the remaining players, if you take any coalition C and then N minus C denotes the remaining players. Now let us say SSN minus C is basically the product of SJ, where J is in N minus C. And similarly SC is the product of the strategies of the players in the coalition C and SN minus C the product of the strategy, the strategy to pulls from the outside that coalition C and this is inside the coalition. Now let us look at, I would like to define this delta SC is basically correlated strategies, players in, so let us say because the players in C are forming a coalition, so therefore I consider all the correlated strategies, similarly delta SN minus C. Now look at Ui of sigma C sigma N minus C, this is the utility that the player i is getting if the players in the coalition C chooses sigma C and people in N minus C choose sigma N minus C. In fact this is nothing but sigma SC in SC Ui SC basically because if they choose SC with probability sigma C SC and SN minus C with probability sigma N minus C SN minus C, so therefore they will get Ui SC SN minus C with the probability sigma C SC into sigma N minus C this thing and sum it over all this thing. So this is the utility that they will get it. Now I can define VC, the worth of this coalition is the best that they can get it that is basically the minimum SN minus C because we need to look at the correlated strategies S sigma N minus C max sigma C then look at the Ui sigma C sigma N minus C, this is the utility that ith player is getting but now you are looking at the sum of all the players in the coalition. Now this I can take it as a worth of the coalition C in the strategic game and then this we can think it as a now a game in a characteristic form. So from the any strategic form or normal form game any N player game you take it from there we can actually introduce a coalition game with transferable utility via for example this representation. So this is not the only way there are other ways of representing we will not go further into this one but we will now start studying the coalition games. Now we will introduce few properties of this TU games. So the first thing is known as a super additive games all these things will be necessary as we go further. So we consider a TU game this is super additive if the following condition holds we if C and D are two coalitions look at the worth of C union D this should be bigger than V of C plus V of D this is true for all C D any two coalitions such that C intersection D is empty. If you take two coalitions C and D non-designed coalition if they form together they are more worth than individually they are so that is the super additive. So let us look at some examples. So let us take N to be 1, 2, 3, 4 so there are four players then V is given by V1, V2, V3, V4 is 0, I am considering only four player game V of 1, 2, V of 1, 3, V of 2, 3, V of 1, 2, 3, 4 this thing. So basically if at least majority are agreeing it then they are getting 300 it is like it is the divide and divide the dollar game and if there are at least two players agree then they are getting the worth 300 if only one person agrees then they are getting 0. So this is the majority voting game. Now another example we can write is that is the following thing take V of 1 is 3, V of 2 is 2, V of 3 is 1, V of 1, 2 is 8, V of 1, 3 is 6.5, V of 2, 3 is 8.2, V 1, 2, 3 is 11.2. In fact this game is introduced in some other context so I only gave here the numbers. So we can verify that this is also a super additive game is super additive. Next I would like to introduce what is called super additive cover. So given game NV the super additive cover of V is the super additive game NW such that V of c is less than equals to W of c for all c contained in N. Basically we are considering you take any game NV and then you find another W another characteristic function W satisfying the super additive structure such that V of c is always less than equals to W c and the W satisfies this previous assumption. So such a game is the super additive cover. In fact how to characterize this I will just mention it but without this thing let us say take let us say Pc basically the set of all partitions of c. What is a partition of c means c can be written as a union of certain sets c1, c2, ck they are all designed and their union is c. So that is basically the partition of c and look at all partitions. Now the Wc is defined as the following way max of j is equals to 1 to k of V of tj such that t1, tk is basically a partition of this. Take any partition t1, t2, tk of c and their union is c. Now look at the sum of V tj's and then you take this maximum over all this thing and define this for every c in this thing in fact then NW is super additive cover. So this is not difficult to prove this one so I will leave it for you to work out. Now we will start introducing some of the important concepts here in the cooperative games. The first thing is imputation. So what is we consider always tu game. So even if I do not mention tu game we are considering only tu games let us look at NW. So imputation is basically is an allocation x1, x2, xn. So you are allocating the worth to each individual satisfying the following thing satisfying 1. So each xi should be bigger than Vi because individually the person is worth Vi so any allocation that you want that should be bigger than xi Vi. This is known as individual rationality. If you specify any allocation where xi is strictly less than Vi he will never accept that because individually he is getting so he will never join in the correlation. So the second thing is that the all the allocations some of all the allocation should be total worth. This is known as a collective rationality because the total worth Vn should be divided among this players. So all this therefore the sum of their individual their allocations should be equal to Vn otherwise you can always increase one of the player. So if you are not considering the collective rationality then they certainly will not accept. So an allocation satisfying these two conditions is known as an imputation and these imputations are the ones which the players will accept. So now let us look at another definition essential and inessential games. So super nv is super additive. So consider a super additive game and this is said to be essential V of i, i belongs to c is less than equals to Vn. You take any correlation c and then summation Vi individually how much they are getting that should always be less than equals to Vn the total worth then you call it as an essential and we call it inessential if the sum of the individual worth that they are getting total how much they are getting if that is exactly same as Vn then this is called inessential because inessential is there is no other way you can allocate in some way. In fact we can prove that if nv is inessential then summation i in c Vi is nothing but V of c it cannot be bigger than this. So this is true for every c in n this is again a trivial exercise to prove this one. For the inessential game only imputation is there is only one imputation available in this thing. So for essential games there are infinitely many imputations. So essential games will have many imputations in fact infinitely many which is not hard to see it. Next we will introduce a strategic equivalence of TU games. In fact even though we have not mentioned it very clearly in the non-cooperative games we can actually say that the two games are strategically equivalent when they are best responsibility. In fact when we were discussing potential games we have introduced this concept. So in fact what you can in general say is that the two games are strategically equivalent if they are best responsibility. Similarly how do you say that two TU games are equivalent, strategically equivalent. So let us look at it nv and n take two TU games they are strategically equivalent if there exists some constant c1, c2, cn of course they are all real numbers and again a number b greater than 0 such that wc is nothing but v of c plus summation ci i belongs to c into b. So if the worth of the correlation c in the second game is simply the worth of correlation c plus some constant which is given by some ci i belongs to c and its entire thing is multiplied by b. So whenever this happens you say that these two games are strategically equivalent. So in a sense the dynamics among the two games are going to be equivalent. So in fact whatever one player gets in one this thing you can actually get a corresponding allocation rule. So in fact as I said the dynamics are going to be equivalent. One important result here is that any super additive essential n person characteristic form game g is basically strategically equivalent to unique game in the players and players we take it then v1, v2, vn all of them individually get 0s and total worth is 1 and 0 less than equals to v of c less than equals to 1 for all c contained in n. The total worth is 1 and then individually they are all getting 0 and then worth of any correlation is in between 0 and 1. So this game is called the 0, 1 normalization of the original game. So in fact this how to prove this one even though we would not go into the proof of this one is looking at this definition somehow you need to choose this constants in such a way that for individual it will be 0 and total worth is 1. So we can in fact it is not very hard to look at this one. So this introduces some important concepts regarding these two games and then we need to now discuss the solution concept which we will do it in the next session. We will stop with this and we will continue in the next session. Thank you.