So, welcome to week 3 of reaction mechanisms, we will just summarize what we did in the last week. So, in week 2 we looked at reaction coordinate diagrams. So, what we had looked at is first of all how to represent reactions as reaction coordinate diagrams and what these reaction coordinate diagrams mean and we had seen how the y-axis gives us information about the stability of the reactants, the intermediates and also gives us information whether the reaction has a tendency to be in equilibrium and also tells us what the rate determining step is. We also looked at what information we can get from the x-axis which is the reaction coordinate. The reaction coordinate corresponds to the geometry of the compound. So, it tells you information about the nature of the transition state and we had looked at the Hammond postulate to say whether the reaction whether the transition state resembles the reactant or the product or intermediate. We had also looked at certain implications and how we can control reactions. So, one example we had seen was of kinetic and thermodynamic control. So, this is where you have a reactant R which can give you a product A or a product B. Now, product A has a higher activation energy but is more stable whereas, product B is less stable and also has a lower activation energy. So, B is formed faster but A is more stable. So, if you do the reaction under kinetic control where speed is important, B is the major product. If you do the reaction under thermodynamic control where stability is important, A is the major product. So, we had looked at examples of these and how you can predict or you can modulate a reaction so that you get kinetic control or thermodynamic control. And we had also looked at the Curtin-Hammett principle. So, this Curtin-Hammett principle essentially tells you the ratio of products formed when you have two conformers A and B. So, A and B are conformers which can interconvert relatively easily. So, that is why you have a small energy of interconversion. The product ratio that is P A and P B is dictated mainly by the activation energy difference for formation of P A from A and P B from B. So, basically this energy difference, I will draw it out so that it is easier for you to see. So, Curtin-Hammett principle essentially tells you that larger this difference greater is the selectivity. And based on Curtin-Hammett principle, this would be the major product. And towards the end of the week, we had just started looking at reaction kinetics. So, we had done an introduction to reaction kinetics because reaction coordinate diagrams do not give you much information about the speed at which the reaction is taking place. So, for that you need to understand the concept of reaction kinetics and I had just introduced you to the concept of the order of a reaction and what a rate law means. So, in today's lecture, what we do is we look at how we derive rate laws. So, let us look at a reaction where A plus B gives product. So, we can measure the change of concentration of A over time experimentally, we can also experimentally measure say the change of concentration of B over time and there is a rate constant that can also be experimentally measured. That is why we call this K observed. Now, one possibility is it is an elementary reaction. So, from what you have studied in the previous class, what is an elementary reaction? A single step reaction. So, if it is an elementary reaction, it means that both A and B are coming together in a single step to give you the product sort of like your SN2 reaction where you have a single step concerted reaction. So, when you have a concerted reaction and let us see both so obviously since it is concerted both molecules would be participating in the transition state. So, the rate law would be, the rate would be given by, so if the transition state is such that one molecule of A comes with one molecule of B, you would see that the reaction would be first order in terms of A and B. So, this would be overall order would be 2 where it is first order in A and first order in B. Now, in this case since it is an elementary reaction, the molecularity of the reaction is also to, it is a bimolecular elementary reaction. So, again I want to emphasize that molecularity term is only used when you talk about elementary reaction and it corresponds to the number of molecules in the transition state. It is typically very difficult to get a term molecular reaction where you have 3 molecules coming together in a transition state but it is not impossible. There are examples where you have a term molecular transition state, it is just that it is not very common. So, now we could derive the rate law quite easily for the elementary reaction. Now, what do we do if the reaction is more complicated, it is deceptive when we are just mixing A and B and getting a product P but when you actually write out the mechanism, it might be possible that A would be going to an intermediate I first and now the intermediate I is what reacts with B to give you P and so each of these processes will have their own rate constants. So, K1 is the rate constant for the forward reaction going to I, K-1 it is the rate constant for the reverse reaction. So, how do we actually derive the rate law for such complex reactions? So, that is what we will do in this lecture. So, when you write the rate law for a reaction like this, by definition the rate law is getting the relationship between the reactants, the rate constant and relating that to the rate. Now, in this case you have multiple processes happening. So, to first write the rate, you always start with the product, so the rate of formation of P would be dependent on the rate constant K2 concentration of I and concentration of B. So, you always start from the right which is formation of the product. Now, this is not the rate law because it does not take into account K1, K-1 and concentration of A which is the other reactant. So, this is not your complete rate law. So, what do we do in complex situations like this? So, a simplification is what is known as a steady state approximation and remember the steady state approximation can be applied only to intermediates or transient and highly reactive species. So, in this case you have the intermediate I. The steady state approximation essentially tells you that after a point in the reaction you have almost a steady concentration of I that means as soon as I is formed it is also consumed. So, initially you have I being formed from A and as soon as it is formed it is consumed to give you P or come back to A. So, this is what steady state means just like the English word you have a steady state of I. Mathematically what that would imply is that you essentially have rate of change of I would be equal to 0, right? So, mathematically it means that after a point you have a steady state that means the rate of I does not change with time as soon as it is formed it is consumed. So, based on steady state approximation the rate of change of I is 0. What that means is the rate at which I is formed is equal to the rate at which I is consumed. Now, let us look at mathematically how we can express both of these. So, what is the process that leads to formation of I? I want you to think about it and write it in your books. If you want you can press the pause button, alright. Now, what are the processes that lead to consumption of I? Again, I would like you to think about it, write it in your notebook and you can press the pause button if you want, alright. So, what leads to formation of I? A going to I is what leads to formation of I? And how do you mathematically represent this? So, the formation rate would be given by K1 times concentration of A. This is a basic application of whatever we had studied earlier, right? When we had looked at just an elementary reaction. Now, consumption what are the processes? There are two processes. One is I going back to A and that would be given by K minus 1 I. And what is the second process? The second process is I interacting with B to give you the product. So, the rate for that would be given by K2 into I into B, right? Because it is dependent on I and B. So, now let us go ahead and with this information. So, we have now the information that Di by dt is given by this, steady state approximation means the rate of formation is equal to the rate of consumption. So, applying steady state approximation we will call it SSA. What you get is K1A which is the rate of formation is equal to K minus 1 I plus K2 I into B. So, essentially what that means is if I were to write the intermediate I in terms of A and B this would be, so now we are taking out I, so it would be. So, in other words K1A by K minus 1 plus K2B would be equal to concentration of I. Now we can go ahead and put that in our original rate equation. So, what you get is the rate would be equal to K2 into concentration of I into concentration of B. So, I will go ahead and write concentration of B and in terms of concentration of I, I would write K1A. So, what you get would be the rate law for this complex reaction. So, now we have expressed the rate in terms of the rate constants and the reactants. You do not see the intermediate here. This is the rate law for this complex reaction shown here. So, we have applied the steady state approximation to get the relationship between the reactants and the rate constants. Now we will look at how to use these equations in understanding reaction mechanisms slightly later. But I hope you get now an idea of how you can apply the steady state approximation to derive rate laws for complex reactions. Now let us consider another scenario. So, I have given you one scenario of A plus B going to P. Now you can also have another possibility where what you have is A and B react together to give you the intermediate I and then I gives you the product P. So, let us try to derive the rate law for this process. So, as I told you, you always start from the right. So, initially the rate you start off in this case it is K2 into I because it is the intermediate that gives you P. Now we need to express I in terms of A and B. So, again we will apply steady state approximation and what you get here would be what are the things which lead to formation of I? So, this leads to formation of I and what are the things which lead to consumption of I? So, this is formation and this is consumption. So, based on this you can say that K1 A B divided by K minus 1 plus K2 would be equal to, so the rate law for this would be K2 K1. So, we are substituting it in this equation. So, this would be the rate law for this complex reaction. So, as you see there is a slight change in the equation depending on the pathway of the reaction. So, here you see the rate law is slightly different from what we had seen in the earlier case which is again different from the elementary reaction. But what you notice here is that in your denominator you do not have any term corresponding to your concentration of B. So, essentially what you can simplify this further to make this K observed A into B. Now what you would notice this is this form is very similar to the elementary reaction although the K observed here has multiple processes of K1, K2, K minus 1 and K2 in the denominator. And experimentally it is very difficult to distinguish between these two because the rate laws have a very similar form. Now, I would like you to derive the rate law for another possibility which I am showing you for the same reaction. So, here you have the possibility of the reactant A going to intermediate I and the side product P1 and the intermediate I then reacts with B to give you the product P2. So, what I would like you to do is I would like you to derive the rate law for this above reaction sequence. So, I would now that we have seen how to derive the rate law I would like you to press the pause button on your video and sit and derive the rate law. So, let us start deriving the rate law together and you can check your answers for those of you who have it in your notebooks. As I said you always start with the last step. So, in this case you have rate is equal to K2 concentration of I, concentration of B because P2 is our final product. Now we will go ahead and apply the steady state approximation. So, by applying the steady state approximation again on the left we can put the processes that lead to formation of I which would be K1 into concentration of A and the processes that lead to the consumption of I, so that would be this is one process which is you have the intermediate and P1 going back to A and then the second process which is so now you can write concentration of I. So, now we will go ahead and substitute that in the rate equation. So, you have the rate is equal to K2 B. So, we are talking about this equation and now for concentration of I I would put K1 A, K-1 B plus K2 B. So, that is K1 K2 AB plus K2 B, so this would be the rate law for the complex reaction that we have seen. So, now what we saw is for a simple transformation which is A plus B going to P depending on the mechanism of the reaction one can have multiple rate laws. So, now what we need to think of is how are these actually useful in determining reaction mechanisms. So, I would like you to think about this particular question. Can reaction kinetics be used to determine reaction mechanisms? I would like you to think about it for 2 minutes. Now that we have done a lot of mathematical simple mathematics, we looked at how a simple A plus B going to P, we can have multiple scenarios and derive different rate laws for this. Now that we have this information, can you use reaction kinetics to determine reaction mechanism? So, I would like you to think about this question and write down the points in your book. You can go ahead and press the pause button. So, what you would see is from reaction kinetics you can actually arrive at mechanisms. So, one as I said we looked at 3, 4 different possibilities. I might not know what the mechanism is. So, looking at the kinetics where I do the experiment determine the kinetics, I can get an idea about the reaction mechanism and we will look at examples of this in the next lecture. You can also evaluate multiple mechanisms. So, I can write say 2 or 3 mechanisms for a particular reaction based on the mechanism I can derive the rate law and doing the experiment I can say whether mechanism 1 is correct, mechanism 2 is correct or mechanism 3 is correct. So, I would need to do a very systematic study where I write the mechanism, derive the rate law and then do the experiment. And a lot of times people also use kinetics to determine a better catalyst for a particular reaction. Since we are not directly looking at this in our particular course, I will not be going into much detail about this third point. In the next lecture what we will do is we will look at how reaction kinetics can be used to determine the mechanisms. So, you can practice writing the rate laws whatever we had done in the previous class, you can practice doing that. So, for A plus B going to P you can again write down the rate laws for each process that we saw. So, thank you and I will see you in the next lecture.


It's a quick home break. You can follow us on Instagram 20 more juice content Thank you so much for following us to subscribe to you every night Today we got a cricket video this is called one in a million moments in cricket One in a million one in a million cricket moments. So I bet one of those moments is somebody gets a Wicked in the butt That would be one in a million moment Here we go. Oh No way He did not expect that And that counts I guess that's a double play What happened oh it went off his leg He can't believe it. Oh He just hit it from afar. It's hard to tell sometimes. Yeah, it happens so fast on the cameras are not there. Oh Nice What happened? Oh, he hit it Like your little celebration Is What what what happened? I don't know it's happening so fast. I can't tell sometimes Okay, there's one pixel on the screen Did I go through his back He just missed it That's an absolute Yeah What in the hell just happened incredible Goes high up in the air Nice love those love those Behind the back Was that a six why didn't even see that when he comes again This time straight up this time longer in the shop slapping it That was fun that always it's always unfortunate because it happens sometimes in like American football and the NFL were like the kickoff will hit the jumbotron right right in the middle It's well just and in baseball sometimes if you're playing in a dome I've that has happened a lot where the ball goes up and it hits the hits the dome. It's always funny. Yep It's it's actually quite funny that even though there's sports are so different There's a lot of stuff that's the same like stuff going into the crowd landing in cups or Catch being made by a fan by a fan or a police officer. Yeah Sometimes in NBA games, they actually catch the players because the players go tumbling into the room They're sitting right there happens also in tennis too Is there another sport that has fans that close as the NBA as the NBA? No going golf Golf people are standing right there in golf.   I Don't know where they sit in relationship to the activity on the field Yeah, I don't either anyways, yeah, that was a lot of fun. I enjoyed that one. I did too That was a good one. We didn't understand Sometimes it happens so fast and I'm just like I did not see what just happened and why And it's interesting because sometimes it seems like the cameras are You know, obviously they're gonna follow the ball But sometimes they don't get to the action as it's happening and I'm not Saying that's an easy task. I imagine I would not be able to follow it either. Oh, no And it's never good when the players like can you show me the rulebook? Where in the where in the rulebook right now and they for the most part at least the sports I know they have to have the rulebook on them. Yeah, if it's questions You have to be able to pull it pull can't just rely on their memory by the way speaking of which Apparently minor league baseball has started something that I just absolutely don't it just why what? Replacing human umpires with electronics. I Think you can have a Little bit of both they already have like you have instant replay That and while you see some of the terrible calls made by some mumps, man I have but that's part of what makes the game the game is this yeah, but that also messes up actor That also messes the majority of the time they get it right the majority of the time But sometimes it's egregious Sometimes because not all the time obviously the NFL which national the NFL is is different I I wish they would go completely. I mean sometimes you never want to because obviously there's ticky-tack stuff that that you don't want called But like sometimes they'll see something and it was clearly and replay I I would say of the three major sports that I watch which are basketball football and baseball baseball by far being the one I watch the most The one that is the most egregious is football As far as bad calls. Yeah. Yeah, just brutal and they have replay, but also That's probably the one that's the hardest to find sometimes though Because there's so much going on. Oh, yeah at all times, but I just there's just one you there's one ball Everything happening is right here usually true, and you've got and I just I can't imagine part of the joy of that game is the umpires and Watching the you know the the moment before the called third strike the moment before the play to first base after the double Playing you watch the umpire to see if the umpire is gonna pound the out or if the column safe The and then arguing a call watching the manager come out and argue and get thrown out of the game You can't what are you gonna do with electronics?  Let us know down below


Welcome back to this NPTEL course on game theory. In the previous session, we proved that Shapley value is the unique allocation rule satisfying certain axioms. Now, in this session, we will discuss another class, another notion of a solution to cooperative games. Known as a nuclear loss, the concept of nucleus depends on what is called excess of a correlation. So let us S is a correlation of N, the excess is defined as the following thing. So E X S is V of S that is the, V S is basically the worth of the correlation S minus how much the people in the correlation S are allocated in this imputation X, here X is a imputation. Recall imputation is defined in the previous lectures, it is basically X is an allocation vector which is both individually rational and collectively rational. So this, of course this we are defining it for any subset S of N. So therefore, there are 2 power N minus 1 X S. So now what is X S? So basically in the allocation vector and their total worth and the deviation from the total worth of that correlation S and this is what. So now this also measures, this measures dissatisfaction with an offered imputation X in a correlation. So in other words what I am trying to say that if S is the correlation that you are doing this V E X S, the X S function here is basically telling you because X is the imputation that is offered to us. So how much dissatisfaction is there? So if V S is minus this thing, if the worth of V S is more that means they are not happy. So you want it as E X S should be as small as possible. So that is basically the idea of this thing. Here the following vector, E X this is basically nothing but E 1 X E 2 X E M X where M is of course nothing but 2 power N minus 1. What is E 1 X? E 1 X is basically E of X X S function corresponding to some correlation S 1 like this E M X is also X S function correspond to some correlation. How are these chosen? S 1, S M are indexed according to the following. The E 1 X is largest X S function E 2 X basically the various correlations look at the maximum excess over all correlations. So that I am writing it as E X S 1 that is E 1 X and the least dissatisfaction is given by the correlation S M for the particular imputation X so that is E M X. So now I have ordered all of them and then I have written this vector E X vector of excesses is in the decreasing order here E 1 X E 2 X E M X this is in a decreasing order. Now to define the concept of this thing I need to introduce some this. So let us take 2 vectors X Y in R M I will say that X is lexicographically smaller than Y if the following thing happens basically this is denoted according to X less than equals to E Y is the notation. What it says is that of course E X the excess vector so there is a small error here this is R N we are taking the 2 allocations X and Y and then I am saying that X is lexicographically smaller than Y this I am using the excess vectors if both are same then of course X is less than equals to Y and Y less than equals to X or there exists some K such that 1 less than equals to K less than equals to M and E I X is same as E I Y for I that is up to K minus 1 and E K X is strictly less than E K Y it is a dictionary ordering so you are looking at it if the first E 1 X E 2 X this thing E 1 X is smaller than E 1 Y then I will say that X is less than equals to Y similarly if X first 2 coordinates are same E 1 X and E 2 X E 1 X and E 1 Y are same look at the next coordinate which is smaller and like that go on. So this is exactly the dictionary order. So now once you define this dictionary order on this E X's that is influencing an order on the allocation vectors X and Y then we take that as this thing. So what now we see is that the lexicographic minimum the lexicographic minimum is called is called the nucleus of the game. So with respect to this ordering if there is an allocation vector X which has the minimum which is the minimum that means every other allocation vector has a bigger access vector than this then I will call that particular allocation such an allocation vector is the nucleus of the game of course does this exist. So in fact the there is a very important result here let me put it this thing there exists a unique nucleus for each cooperative game. So the theorem actually says that given any cooperative game and we there is always a nucleus in fact it is unique. So now let us compare with the three notions that we have seen so far. The first we have seen the core we know that core can be multiple the size of the core can be as large as possible. The second is Shapley value which is unique which is actually a very famous concept that is used in the cooperative games Shapley value is uniquely characterized and now we are seeing the nucleus once again this is a unique this. So nucleus and cooperative game enjoys the uniqueness whereas the core need not be unique. So now first we will work out some examples and then we will go to the other parts. So let us look at an example. So let us say a small company goes bankrupt owing money 3 credits. So for a let us say some 10,000 not specific units for B 20,000 and for C let us say 30,000 but company has 36,000 only basically to cover this. So how should the money be divided? So let us look at this particular example. So for A they have to give 10,000 for B 20 is for C 30 and but the company has only 36,000. Now how this money be divided? So there are several ways you can do it for example you can say the pro rata why can do it but now we will actually work out how this nuclear less and for example one Shapley value help here. So let us write down the characteristic function there are 3 players. So these are the 3 players and then characteristic function V of empty set of course is 0 and V of ABC is 36. So measured in 1000 so let me not worry about that and so let us say if V of A what will be V of A? If V of A is 0 the reason is that B and 6 gets 20 and 30. So if A alone is coming that B and C can together climb the money and then they can try to split therefore V of A becomes 0. Similarly V of B will also be 0 because A and C their climbs is bigger than 36. So therefore V of B will also be 0. So V of C if you look at it if A and B form a coalition if I divide the if they take their money then 30,000 the remaining 6. So therefore this will be 6. Similarly we can find that if we write down in the same fashion V of AB will be 6, V of AC will be 16, V of BC will be 26. So the same way we can argue and then this thing. So now this is basically the coalition form game now let us take an allocation X to be X1, X2, X3. Let us say this is a efficient allocation. Recall what is an efficiency X1 plus X2 plus X3 has to be 36. Now we will look at the X's functions. So we have to write down here S V S E X S. So if I take A as this thing V of S is 0 if A is the coalition S is equals to A then V of A is 0 and E of X comma A because V of S is 0 this will be simply minus X1. So B if I take it 0 this is minus X2. So like that I can actually calculate all the things C for C it is 6 it is 6 minus X3 AB this is 6. Therefore this will be 6 minus X1 minus X2 AC is 16 this is going to be 16 minus X1 minus X3 BC is 26 and then this is going to be 26 minus X2 minus X3. Then these are the excess functions that we have defined. Now what we really need to now look at is for different X's allocation vectors what you have to calculate these values for example if I take X to be the 6, 12, 18 we can calculate this to be minus 6 minus 12 minus 12 this is also minus 12 this will be minus 8 minus 4. So like that we can do it for for example 5, 12, 19 if I take it we can calculate this as minus 5 minus 12 minus 13 minus 11 minus 8 minus 5. Like that we can do it for many this thing. In fact by looking at these things and by little work we can say that the nucleolus here is going to be 5, 10.5, 20.5. So it requires one to calculate this because it you remember we have to for each X we have to look at the order them and then try it. So here it is worked out for two examples but one has to be little careful to see what exactly is the nucleolus. So it requires a little work but we can do it. Now how does the Shapley value here? In fact we can say here is that Shapley value is going to be 6 here, Shapley value for the B is going to be 11, Shapley value for C is going to be 19. Just apply the formula and then they use this thing and what you have here is the allocation vector here is 6, 11, 19. So in fact if you try to compare these two examples we can easily see that these two are not same. So this is a very important point to observe here. Now we will try to give some ideas of the proof of this nucleolus in existence of nucleolus. Let us say look at the proof. Of course we will give only some ideas of existence. So let us first remember the following thing. So the components what is E1X? E1X is basically the maximum of all the excess functions. So it is a maximum of all I1 to M recall M is 2 power N minus 1 of the excess functions, maximum of all those things. Now what is E2X? E2X is the second max of this thing. So in fact it can be written in the following way minimum J is equals to 1 to M of max I not equals to J E X S I or E I we can put it. So let me write this is nothing but E I X S I or E X S I. So you take pick any J and look at the maximum of all the I's such that I not equals to J. So for example to illustrate this one let us take J is equals to 1. So what is this one? Max of I not equals to 1 E X S I that means exclude E X S 1 and take the maximum of the remaining N minus 1. Now suppose look at this one. Suppose the one corresponds to the maximum of that E1X let us say then here this will be the second maximum because E X S 1 is removed and maximum among them. Now in all the other things E X S 1 will appear because when you take J to be different from 1 then 2, 3 and the 1 is always there so those appear here. So therefore the when I take the minimum of them this is nothing but the second max among the excess functions. Now like that E 3 X will be the minimum of J K. Now we take 3 indices and then you take J different from K and then take the max I not equals to J not equals to K of the E X S I. Now because you take 2 at a time you are removing 2 of them and then looking at the maximum. Now if both J and K are exactly the first max and second max then that will give you the minimum among this one and then like that. Now this way you can write all the things. Now the excess functions now clearly can be written in this form. Now most important thing to note here is that these excess functions E 1 X E 2 X E 3 X these are all continuous functions. So all of these are continuous functions. So therefore the maxima and minima the reason is set of all imputations if you really go back and see it because they are efficient and other things they are positive and all they are that is a set of imputations is a compact set of all the allocation vectors because every person is satisfying individual rationality they are above something and then the sum of them is less than equals to V n. Therefore the imputations are always the set of imputations is always a compact set. Now these are all continuous functions. So therefore every function all these E 1, E 2, E X they enjoy this existence of maxima as well as minimum that we have used it earlier also. So therefore okay now E 1 X for example let us say look at the E 1 X let us say it attains minimum value. So E 1 X attains its minimum value because E 1 X over set of all allocations X is a continuous function over a compact set. So therefore it will have a minimum value. Now that means E 1 X at this minimum wherever this minimum is achieved that is going to be the point where the excess function can have the minimum value. In fact whenever such if the minimizer is unique then this minimizer is going to be nuclear less. So there is nothing else to be done when you have such a minimizer here but the problem is that this minimizer need not be unique there may be a set of minimizers. So once you know that there is some set of minimizers what you need to go here is that look at the next level E 2 X okay and then E 2 X is again a continuous function. Now look at the set of all minimizers that E 1 X is minimized and then on that particular things then look at the second function E 2 that also will have a minimizer and go on. So once for example if U 2 X has a unique minimum then automatically that allocation is now fixed. Of course remember when I go for minimizing E 2 I only need to look at the set of minimizers of E 1 and therefore E 1 is already minimized that means E 1 X will be same for any 2 allocations in the set of minimizers and then look at E 2 if the minimizer of E 2 over that is unique then that gives you the nuclear less if the set of minimizers are not unique then I go to E 3 and then go on one by one. Now in this fashion if you really go it there is a some set where all these things are minimized therefore this idea gives existence. Now the problem here is that how about uniqueness. So in fact the uniqueness can actually be proved but it is slightly tricky we will not go into the details but the idea of this uniqueness is that you have this set of minimizers as a convex sets and then the intersection of all these convex somehow that they become really unique. So I will not go into the details but the details can be for example found in the original paper of Schmidler. Schmidler is the one who developed this notion of a nuclear less. So original paper of Schmidler or in fact we can look at for example Schmiedler's book of game theory. In fact this is the book the material from this book is used heavily in this course. So now of course we will not discuss about the uniqueness but existence is proved. Now note that before we go further I would like to make a point here that the cooperative games we have multiple notions we are defining it so and particularly the nucleus and Shapley value are unique but then if you look at it the computing nucleus is harder than computing the Shapley value because Shapley value is given by a formula and computing nucleus is a little harder. Now there is one very interesting question that people ask here is that are there some learning methodologies or some non-cooperative framework to these cooperative solutions. So this is a very interesting problem in fact there is a lot of work happening but we will not touch upon this but in fact a good point to look at it is to start with this Nair Harri's book or even Maier-Sens book on game theory. So with this we will stop in the next session we will consider another problem from commutative games.


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For the rich taste of fine tobacco for smoothness and mildness, remember, there's never a rough puff in a Lucky Strike. The Lucky Strike program starring Jack Benny with Mary Livingston Phil Harris, Rochester, Tennessee, and yours truly, Donald. Ladies and gentlemen, on the corner of CBS Square, there's a small drug store which is frequently visited by radio personalities. At the moment, Mervyn, the soda jerker, is talking to Flossy, the new waitress. Take it easy, Flossy. Don't be so nervous. I can't help it, Mervyn. After all, this is my first day as a waitress. It's nothing to worry about. What did you used to do? I was in the movies. You? You was in the movies? Yeah. I was head popcorn popper at the Pantages. Well, why did you quit? My boyfriend got tired of kissing my salty fingers. See, I hope I make good at this job. Oh, don't worry. You'll do all right. Uh-oh, get ready for business. Here comes Jack Benny and some of his cash. Oh, gee, celebrities. Who are they? Dennis Day, Don Wilson, Phil Harris, and Jack Benny. Hey, see, they're starting across the street. Oh, yeah. Which one is Jack Benny? The one that looks like he won't make it. Hey, you better set a table for him. OK, Mervyn. Give me four glasses of water. You'll only need three. One of them never touches a stuff. Now, come on. You better get busy, Flossy. Now, come on, fellas. There are plenty of tables. Let's sit over by the window. OK, by me. I am, Mervyn. Good health to all from Rex Hall. Phil, why do you always make him do that? Well, it only cost me a buck a month. And if my sponsor comes in, he'll be impressed. Why? Because this is a thrifty drug store. Well, let's sit down, fellas. Your audit, please. Well, ha-ha-ha-ha-ha. Ah. Get some new talent here. Your audit, please, gentlemen. I'll take a ham sandwich on Ryan, a cup of coffee. Yes, sir. I'll have the same. Yes, sir. And I'll have a glass of milk and some apple pie on a mode. Yes, sir. What flavor ice cream do you want on your pie? Oh, no, ice cream. Just put on a scoop of mashed potatoes. Dennis. Dennis, why in the world do you order pie with a scoop of mashed potatoes? Because I'm a slow eater. What's that got to do with it? Ice cream melts. I asked him, he answered me. Now I'm too sick to eat. What do you have, sir? Well, I'll have a shrimp cocktail, some clam, chowder, a porterhouse steak, hearts of lettuce with Thousand Island dressing, some green peas, french fried potatoes, a piece of chocolate layer cake, and a cup of coffee. Can you remember that? I hope I can carry it. That'll be all, Miss. Thank you. I'll be right back with your orders, gentlemen. Hey, fellas, hey, she's a kind of cute-looking girl, isn't she? Yeah, uh, right, personality. Beautiful red hair, sunny smile, trim ankles, cute little figure, and they'll, Phil, remember, you're married. I know, but I can still take inventory, can I? Yes, yes. Hey, Dennis, what are you looking at? There's lipstick on my glass. Lipstick? Well, I'll call the waitress and tell her. Oh, no, no, no, this is exciting. Dennis, what's exciting about lipstick on a glass? It's spring, Dad, spring. Oh, be quiet. Say, Jack, there's a new record here in the jukebox called Deary. It's sung by Dennis. By me? Oh, boy, I'm going over and play it. All right, go, go. Nobody touched my glass. We won't. We won't touch it. Play the record, will you? Pulse to the Sousa band, my wasn't the music grand. Chatter parties down by the seashore. Every 4th of July, test your memory, my deary. Do you recall when you couldn't even fix a running boat under a chandler's seat? Deary, life was cheery. In the good old days gone by. Deary, do you remember when most gents owned a sparrows? Wing collars and derby hats. Lovely ladies swingin' their bustles. May those gentlemen si-test your memory, my deary. Do you remember watching John L. win every fight? Because nobody ducked from Sullivan's right, dearie. Life was cheery. In the good old Jackie. Do you remember watching in Illinois? Gosh, weren't you just a boy that was long before love and bloom? And maybe Brahms lullaby, test your memory, my Jackie. Can you remember with very scant when you played a lady in Charlie's aunt? Jackie, life was cheery. In the good older than I. I'm 39. Will you gentlemen have anything else? No, not me. I've had enough. I'll bring you a check in a minute. Hey, say, Phil, how about coming over to my house for a little gin rummy? Well, I like to, Jackson, but I got to go home and wash Alice's hair. Phil, you wash Alice's hair? I used to be on for fetch. I got to use that stuff up some way.  I'll wash that too. Never mind. Say, Don, how about you coming over to my house? Well, I wish I could, Jack, but I've got to go back to the studio and make some spot announcements. Oh, say, Dennis, would you like to come over to the house, play a little gin rummy with me? No. Why not? You cheat. I cheat? I've been watching you. When you play, you deal off the top of the deck. What? The man I played with on the train all was dealt off the bottom. Heaven's sake, kid, when will you learn? Check it, Jack, gentlemen. Oh, I'll take it. Good, good. Well, fellas, I'm going home. Mr. Benny, do you want me to drive you? No, thanks, kid. It's such a nice evening I think I'll walk. So on, boys. So on, Jack. See the weather's bomby. Spring is the nicest time of the year. Trees are green, flowers are blue. Makes a fella feel good. Toot, toot, toot, sing goodbye. Nuh. Toot, toot, sing. Where does Jolson get off saying he doesn't go nyeh? Da, da, da, dum, bum, bum, bum. Excuse me, mister. Huh? Will you spare a dime for a cup of coffee? A dime? Well, let's see. I haven't got a dime. The smallest I have is a half a dollar. I haven't eaten since yesterday. Well, look, look, buddy, here. Here, take the half a dollar. Gee, mister, thanks. Thanks a lot. You're welcome. You're welcome. Gee, that was nice of me. He only asked me for a dime, and I gave him a half a dollar. I wonder if it's deductible. What's the difference? Toot, toot, toot, sing goodbye. Half, half dollar, goodbye. Da, dee, da, dum, bum, dee, dum. Da, da, da, dum, dum, dee, bum, bum, bum, bum, bum, dee, bum. Hello, Mr. Benny. Well, Mr. Kitzel. Who was your friend, too, who I just saw you talking to? My friend? Oh, oh, that was some poor man who wanted a dime for a cup of coffee, but I gave him a half a dollar. Bless you, huh? Thank you. By the way, Mr. Kitzel, where are you going? I'm on my way to the baseball game. Oh, yes, yes, there is a game tonight. The Angels and San Diego. I didn't know you were such a fan. Oh, my. But I'll tell you one thing, Mr. Benny. The baseball players are not like they used to be. You're right. Some of those old timers were really great. Oh, my. When I think of such a players like Christy Matheson, Ty Kaap, Rabbi Marenville. No, no, no, Mr. Kitzel. You mean Rabbit, Rabbit Marenville. I have a team I never heard of. Oh. Well, I don't want to be late, Mr. Benny. See you again. Oh, by the way, Mr. Kitzel, while I was in the drugstore, I meant to buy a package of Lucky Strikes and I forgot to. Do you happen to have a... Of course, of course, of course. I smoke Lucky Strikes. Oh. Yeah, I have one of mine. Well, thanks, thanks. Believe me, Mr. Benny, I've been smoking Lucky Strikes for an eye onto 25 years. Your hair? Yeah. I'll never forget the first time I ever smoked a Lucky. The mildness, the smoothness. Ho, ho, ho. You really like them, eh, Mr. Kitzel? Like them. Not only are they round and flim and fully packed, but there's never a refraft in a Lucky. No, no, no. You mean rough puffs and refraft, rough, poor, flight-warning, enjoying. Thank you. Well, goodbye, Mr. Kitzel. See you in a minute. See, goodbye. Yeah, Mr. Kitzel is a nice guy. But then I'm a nice guy, too. Yes, he'll only ask me for a dime and I gave him 50 cents. Yeah, I wish Luella had been there to see it. Maybe I'll call her when I get home. Well, here's my house. Wasn't such a long walk after all. See now, which one is my door key? Here's the key to my car, the key to the garage, the key to my hope chest. If Mary doesn't ask me soon, I'm going to start wearing those things. Oh, here's the key to the door. Rochester, I didn't know you were home. I'm in the kitchen. Rochester, isn't this your day off? Yeah, but I thought I'd stay home and get caught up with these dishes. Well, look, you've let a whole week's dishes accumulate. Why do you let them pile up like that? It is my fault it's that new soap you bought. We just can't get together. What do you mean you and the soap can't get together? When tide's in, I'm out. Oh, say Rochester, on the way home... Excuse me, boss, I want to put these clean dishes away. Rochester, you know, on my way home, some poor fella asked me for a dime. Uh-huh. But I gave him 50 cents. Rochester, why'd you drop those dishes? All I said was I gave a man 50 cents. Rochester, you didn't have to push that second stack off the drain board. I didn't touch them. They jumped off by themselves. What a mess. Boss, look at me. Huh?  I certainly did, Rochester. You know, if I had known the wonderful feeling, the warm glow I'd get from being generous, I would have started earlier in life. Boss, would you like to talk about my salary? Manana. The word I picked up in Palm Springs. Rochester, I've had a long day, so I think I'll go to bed. Okay, good night, boss. Good night. T-t-t-see, goodbye. T-t-t-t-see, don't cry. Da-dee-da-dum-boom-boom. Da-da-da-dum-boom. Yeah, I bet I'll really sleep tonight. Ah, this bed feels good. Feel good to get my shoes off, too. That's a funny-looking sock. Oh, for heaven's sake, I told Rochester a million times, don't put my gloves in that drawer. No wonder my toes were cold. They were separated. They're my pajamas. Oh, here they are. I don't ever remember being as tired. Oh, my goodness, I almost forgot. Dear diary. April 9th, 1950 cents. I mean, 19. Today, I did a wonderful thing. A needy person asked me for a dime for a cup of coffee, and I gave him 50 cents. Rochester, what happened in the kitchen? I don't know. I'm in bed. Well, I better get to bed, too. I got to get up early tomorrow morning and play golf with Mary. Oh, dear, I forgot to call the well until they're about giving that man 50 cents. Yeah, but then maybe that's too hammy. Uh, it was nice of me, though. I didn't even know them, fella. They gave him 50 cents. 50 cents. He didn't have a gun or anything. I just gave him 50 cents. This is your CBS Worldwide correspondent bringing you the event you've all been waiting for. Today, dignitaries from the four corners of the earth have gathered at this banquet to pay homage to the most generous man in the world, Jack Benny. That's me. He's talking about me. They're giving me a banquet. Yes, ladies and gentlemen, this great event is coming to you from the banquet room of the Taj Mahal. The dais is replete with dignitaries. A hush falls over the audiences. The master of ceremonies rises to his feet. He is none other than the honorable Winston Churchill. Winston Churchill at a banquet for me. Ladies and gentlemen, it is with a deep feeling of pride that I have come here to preside on this wonderful occasion. Little did our guest of honor know that his exhibition of generosity generosity would shake the very foundation of the world. I know that most of you were as shocked as I was. When the news reached me, I was sitting on the front steps of number 10 Downing Street. As you all know, I'm not quite inside yet.  James Cagney. How do you do, ladies and gentlemen? It's a pleasure to be here. There are so many things going to be said tonight about Mr. Benny tossing four-pitch to a panhandler that I, well, that is, well, I'm not the kind of a guy who makes faraway speeches. But I'd like to say one thing. Good health to all from the Ties Mahal. And now for our next speaker, another celebrity who not only regards Mr. Benny as his best friend, but has for many years had the good fortune of being Jackson's next-door neighbor. He called me Jackson. Winnie called me Jackson. And here he is, our next speaker, Mr. Ronald Coleman. Ah, thank you, ladies and gentlemen. If I were king, I couldn't possibly wish for a better neighbor. If I were king. Yes, if I were king, I would bestow upon our guest of honor the title of knighthood. If I were king. If I were king, I'd see that every man in my kingdom had five acres of land, ten room house, and a glass with lipstick on it. And now, before I conclude, I would like to assure Mr. Churchill that the 50 cents that Mr. Benny gave away came out of his own pocket and will not be deducted from the Marshall plan. Of course, sir. Thank you, Ronnie. That's quite all right, Winnie. That's quite all right, Winnie. Ladies and gentlemen, while Mr. Churchill is introducing the next speaker, I want to take this opportunity of passing through this vast audience and pointing out some of the other celebrities who are here today. At the far end of this table, I see Princess Elizabeth. Seated next to her is Madame Shang Kai-shek. Looking down the table, we see Queen Juliana of the Netherlands. And seated next to the queen, we have a lovely lady dressed in ermine cape, star sapphire tiara and a diamond bracelet. I beg your pardon, Miss, but I've taken the liberty of describing your jewels, so now may I ask, who are you? I'm the girl who works in the drugstooler. Oh, well, who is this sitting beside you? John Wilson. He hasn't finished eating yet. So now, once again, we take you to the bayers and Winston Churchill. And now, ladies and gentlemen, another great man and speaker, Mr. Carrie Grant. Good evening. Carrie Grant is here, too. Good evening, ladies and gentlemen. I'm happy to be here tonight at the Taj Mahal honoring this great man. I'd like to say only this. Mr. Benny may have had the experience of being Charlie's aunt, but I was a male war guard. Thank you, Mr. Grant. And now we come to one of the highlights of the evening. A man to whom Mr. Benny gave the 50 cents. Mr. John Alice C. Soroni. That's the guy. That's the guy he gave the money to. Ladies and gentlemen, I'm very happy to have this chance to talk to you because, well, I have a confession to make. When Mr. Benny gave me the 50 cents, I don't want to buy nothing to eat. I just want to take the 50 cents and buy a sweepstake ticket. Sweepstake ticket? Well, I did. And that night I went home and I was just hanging around the house. I wasn't doing anything. Just hanging around the house. I didn't feel like doing anything. So I was hanging around the house and... I started listening to the radio. All of a sudden, the radio started talking to me. And I had the radio for three years. I never said anything to the radio. The radio never said anything to me. I didn't speak to any furniture in my house at all. All of a sudden, the radio said to me, I don't speak to any furniture in my house at all. All of a sudden, the radio said, did you notice you had a window of $150,000? I said, who? You. Me? Yeah. I said, do you know what the number is? He said, your number is<PHONE_NUMBER>-M69907.<PHONE_NUMBER>, I said, holy smoke, that's me. $50,000. $150,000. That's what I want. $150,000 is mine. I gave you the 50 cents. I paid for that ticket. It's mine. It's mine. I paid for it. I paid for it, do you hear? It's mine. I paid for it. Boss, wake up!  Oh. Oh, yes, I was dreaming. What were you dreaming, boss? About giving that man 50 cents.  Scientific tests prove lucky strike is milder than any other principal brand of cigarettes. Yes, scientific tests prove lucky strike is milder than any other principal brand of cigarettes. Confirming these scientific tests, an independent consulting laboratory with more than 15 years experience in cigarette research reports lucky strike, mildest of six major brands tested. LS, MFT, LS, MFT. Lucky strike means fine tobacco. That's why with every lucky, you get the rich taste of fine, light, naturally mild tobacco. Yes, smoothness and mildness with never a rough puff. Just listen to the words of Mr. Raymond W. Crutchfield for 16 years an independent tobacco warehouseman from Reedsville, North Carolina. Recently, he said, in my experience as a warehouseman, year after year, I've seen the makers of lucky strike by good smoking tobacco. Fine, ripe, mild leaf that makes one great smoke. I've smoked luckies for 20 years. So smoke a lucky. Prove to yourself what scientific tests prove. Lucky strike is milder than any other principal brand of cigarettes. Let your own taste and throat be the judge for the rich taste of fine tobacco. For smoothness and mildness, remember, there's never a rough puff in a lucky strike. So round, so firm, so fully packed, so free and easy on the draw. Ladies and gentlemen, the impersonations of Winston Churchill, James Cagney, Cary Grant, and Mr. Silvone, the man I gave the 50 cents to, were done. All of these impersonations were done by Frankie Fontaine. Ronald Coleman was impersonated by Dennis Day. The impersonation of giving away 50 cents was done by Jack Benny. Thank you, John. Thank you. Good night. Put them out. Take every precaution you can to prevent fires. Be sure to hear Dennis Day in a day in the life of Dennis Day. Stay tuned. The Amish 90's shall be followed immediately. Frankie Fontaine appeared for the courtesy of 20 cents per clock, produces a technical production, cheaper by the dozen, starring Clifton Welles. This is CBS, the Columbia Broadcasting System.