 Welcome back, we were talking about expected utility theory and I ended with its main premise which is that one has a way of comparing probability distributions or lotteries on outcomes and the presumption was that if you had this then this was actually encoded in terms of a utility function the expected utility and what one had to do was look at the expected utility of the outcome under a particular decision and look for the decision that maximizes this expected utility. So, let us actually fructify this in terms of an example. Let us look at this more concretely in terms of an example. So, here is an example from an investment example. So, the problem here is of allocating our capital we have a capital of we have a capital of say you know just one dollar for simplicity we have a capital of one dollar and we have two investment alternatives to investment let us call these alternatives A and B. Now, the investment option A what option A does is option A yields alternative A gives you for every dollar one dollar that you invest it actually gives you 1.5 dollars with certainty. So, the outcome that it results is the is only one particular outcome it gives you 1.5 dollars with certainty. Now, alternative B has is actually probabilistic. So, alternative B will give you with probability half it was going to give you 3 dollars with probability half and it is going to give you 1 dollar also with probability half. So, with probability half it will keep your investment the same it will keep it at it will not erode your investment it you will still get back one dollar the one dollar that you invested and with probability half it will triple your investment. On the other hand A is guaranteed to give you 50 percent to give you 50 percent return it will grow your investment to 1.5 dollars with certainty it is guaranteed to do so. Now, in our framework therefore what we need to decide is what fraction out of this 1 dollar should you invest in A and what fraction should you be investing in B. So, the fraction let us call this fraction D. D in the interval 0 1 is the fraction to be invested in A. So, therefore our decision then is the fraction that we are investing in A the rest is going to be invested in B the set of decisions then is this interval 0 to 1. So, we have to choose a decision from this set from this set of decisions. What is the set of outcomes that can arise well we can we can see here the outcome can be anywhere from the range 1 to 1 to 3 depending on the fraction that we invest you could get all the way up to if you invest all your money in 3 in B and it turns out that you that the investment actually triples then in that case you would get you would get 3 dollars. So, if you put all the entire 1 dollar in B then you would get 3 dollars if you put if you but then there is also a possibility that you will get only 1 dollar. So, depending on the fraction that you invest you could get any anything in the range from 1 to 3. So, the outcome space O is this interval from 1 to 3 the actual outcome the actual outcome is a function of the state of what actually comes out of the investment in B whether this state of the world gets realized or that state of the world gets realized. So, this state of the world let us denote this one here by the this state of the world here let us let me denote this by by omega 1 here and let me denote this by omega 2. So, the states of the world then is capital omega which is these two possibilities omega 1 and omega 2. So, the actual outcome then is is going to be f of D, omega. So, if I invest a fraction D in A then I am going to get 1.5 dollars times D from A because whatever fraction I invest for every dollar I invest it is going to give me 1.5 dollars. So, if I invest D dollars or D fraction of 1 dollar it will get 1.5 D as my return on it plus whatever the remaining is invested in B. So, the rest of this term is going to depend on which state of the world gets realized. So, what we will do is instead of writing it this way I will write this as follows. So, I will get 1.5 D plus 3 into 1 minus D why is this 3 into 1 minus D well it is 3 into 1 minus D because I will get 3 dollars for the 1 minus D that I have dollars that I have invested in alternative B when omega 2 gets realized. So, when omega 2 sorry when omega 1 gets realized when omega 1 gets realized. So, this will happen with probability half alternate in the other case when omega 2 gets realized I will get 1.5 D. Remember I still get the 1.5 D from investment in A. So, I still get 1.5 D plus 1 minus D this happens in scenario omega 2 and that happens with probability also happens with probability half. So, this is therefore the actual outcome that will get realized. So, obviously this is a function of both the state of nature as well as the decision or the fraction that you are going to invest. So, let us just look at a few numbers how this behaves. Now notice that if you see if you let us look at if one possible thing to do is again although I have already pointed out the fallacy unit let us look at the average rate of return. What is the average rate of return in both of these investments the on average B is going to give you is going to give you an average of 3 and 1. So, on average B is actually going to give you 2. So, if you put the entire 1 dollar in B on average you are going to get 2 dollars and on and whereas A with certainty gives you 1.5 dollar if you put the entire 1 dollar in A. So, if you put the entire 1 dollar in B you would on average get more than what you would get in A. So, consequently you might think one may assume that well you would want to put all your money in B, but if you ask most people most people will not behave in this way this is not how this is not really their attitude towards lotteries and towards uncertainty and towards risk. At the same time if you look at if you look at alternative A now alternative A gives you 1.5 with certainty and that is better than the worst case outcome of B. So, the worst if you just think only of the worst case the worst thing that could happen if you put your money in B if you put any amount of money in B. If you think of the worst that could happen the worst that could happen is that whatever money you put in B does not grow at all it you know you put in you put in an amount 1 minus D in B and it remains at 1 minus D. In which case you would have been better of putting all that money actually in A itself. So, the worst case the decision that that maximizes your worst case that gives you the most money in the worst case is the decision to put all the money in A. So, to make this more concrete so if one simply if one looks at so the worst the way to think about the worst case was one looks at the worst possible scenario over all possible states of the world that could arise of this and you try you max you look at the decision that maximizes the worst case the or your return in the worst case. Now in the worst case this actually you will see this is nothing but maximum over D in I am just putting capital D as 0 1. So, I am replicating the terms from the previous thing. So, it is from the previous slide that is that these two terms these two terms here. So, one looks at this and one looks at the minimum out of these two which is one point minimum of 1.5 D plus 1 minus D and 1.5 D plus 3 times 1 minus D. So, obviously if you compare these two it is clear that the first term here is the lesser of the two and then what we are doing therefore is maximizing choosing the D that maximizes the first term and so that is nothing but and this what does this say well this says simply that one should put all one's money in in alternative A. So, the maximum of this is 1.5 itself and that comes with D equal to D star equals 1 in other words the entire money should be invested in A. So, the worst case so, the worst case outcome if one looks at the worst case outcome the answer is invest everything in A. If you look at only the average outcome the answer is well that what that tells you is invest only in B. So, what expected utility theory is basically telling you is not do is to not do any of this what it tells you is in fact what you should be looking at is the utility of the outcome and then find that value of D that maximizes the expected utility. So, what the what expected utility theory instead tells you to do it tells you to do is to maximize this the expected utility of the now there are two possible two possible values of omega. So, this expectation as I said this expectation is over omega and there are two possible values of omega and what they will result in is those two expressions that we wrote on the previous slide. So, what you are doing therefore is maximizing. So, if I am just evaluating this expression I have the probability of so, I have the probability of omega equal to omega 1 which is which is half and the utility that comes from omega equal to omega 1 when omega is omega 1 the outcome is 1.5 D plus 3 times 1 minus D and so, what I have is the utility of this particular outcome I have U of 1.5 D plus 3 times 1 minus D plus half again times U of 1.5 D plus 1 minus D. So, what you expected utility theory is telling you is to choose a D that maximizes this function it is not tell it is not saying you look at the worst case is not saying you look at the average case or any of that it just says look at this particular function. Now, obviously the question arises where are you going to get this function from is there and who is going to tell you this function who is going to tell you this function U. So, the important most important sort of contribution or result of expected utility theory is that there is in fact such a function once you have a preference ordering on the set of lotteries this there always exist such a function. And so, this function is and is to a large extent unique there is a you can say without almost without loss of generality that is this function is unique and it is basically capturing for you or your attitude that is encoded in the preference ordering that you have in the on the set of lotteries. So, now what we will do now before I come to that particular theorem which gives guarantees the existence of the utility function let us actually do one example. Let us take one particular utility function and let us actually work this out work out what this actually tells you. So, what we will do is we will take U of O to be equal to say alpha O minus O square. So, when an outcome O arises the utility that you get from that outcome is some is alpha O minus O square and where alpha is some scalar I will we can fix some is a scalar whose value we will fix. So, the utility must have has needs to have a certain set of properties. So, for example, we need for instance that the utility has to be increasing. So, more money is gives us more utility. So, the utility will be increasing for all of this to hold what we will impose is that alpha is actually strictly greater than 6. So, you can verify that when alpha is strictly greater than 6 U is actually increasing on the set of outcomes on the space 1 to 3. Now what we will do is now we can now plug this utility in into this into this expression that we had into this particular expression and plugging in that particular plugging that in we can then find the maximizing D. So, it turns out that the optimal D star then the maximizing D star is some is takes this form it is 0 with if alpha is greater than equal to 8 and it is 8 minus alpha divided by 5 if alpha is between 6 and 8. So, if alpha is between 6 and 8 it is basically you notice that this term D star is actually is neither 0 nor 1. So, consequently what it is telling you is that one should not actually be investing the entire amount in A. So, this is effectively what it is capturing our intuition about how one would make a decision in this kind of a manner. We would not put all your money in A nor would you want to put all your money in B what you would want to do is divided so that some amount is in A and so that you get your fixed return as well and some amount is in B. So, that you also get you get you get the benefit of some additional returns that that that are provided because of the probabilistic outcome of B. So, but you do not want to go to either extreme where you want to put everything in A or everything in B. So, this is what it is getting what is getting captured for alpha between 6 and 8. When alpha is very much alpha is greater than 8 in that in that case effective if you see what seems to be happening is that you know the function here is becomes more and more towards linear as alpha increases. So, if the function is more and more towards linear effectively what this function is trying what it is what the utility is trying sort of trying to suggest is for you to look at is to ignore is to ignore the second term is to ignore the theta square term and look at only the first term in which case your boss what you are effectively doing is look at looking at therefore the expected outcome itself. And as a result when alpha is greater than equal to 8 your decision begins to more mimic the what you would do if you were to only look at the expected outcome and when you were looking only looking at the expected outcome what were you doing well you were putting all your money in B. So, that is what is getting reflected in this. So, when alpha is greater than equal to 8 at that for that value your D star is 0. So, you do not put any money in A at all you put everything in B. So, your attitude towards risk is actually getting captured here by the shape of the utility function. So, when your alpha is when your alpha is large your utility function on the interval 1 to 3 tends to be a little more like a linear function for large alpha for alpha small it tends to be a little more like a function like this. So, the shape of this utility function is capturing the way the is sort of encoding in it your attitude towards risk. So, if your alpha is large your you tend to behave more like your utility function behaves almost like a linear function and then therefore and therefore it only emphasizes for you the first moment of the first moment of the outcome which is therefore the mean of the outcome. So, here is one key this is actually a fairly interesting point when alpha is when only the first moment is mattering the first moment is mattering only when alpha becomes large and when alpha becomes large your utility becomes almost linear. In general though all moments of the utility will matter. In this case the utility is quadratic. So, both moments the first and the second moment would matter. If this utility was not quadratic but say some other concave increasing function then potentially any all the moments would begin to of the uncertainty of the outcome would begin to matter. So, essentially so if you have a general utility function which is you know which is concave, differentiable and so on concave and differentiable the maximizing D would have in it the role of all the moments of uncertainty. So, all moments of the outcome would in general matter. So, for example, you can take say for a utility for example, which is e to the minus lambda times theta lambda times O 1 minus e to the minus lambda times O this sort of a function is has this sort of shape and we can see when we expand the exponential you can see that every it actually expands as a power series of as a power series in theta and in that case all it is not just theta or theta it is not just theta square but theta cube theta raise to 4 etcetera. So, essentially every possible moment of the outcome has a role to play. So, when the utility is linear you are it is you only care about the first moment. So, this intuition what this is also telling you is that this particular attitude of looking at only the average outcome is essentially born out of an underlying thinking underlying assumption or a mental attitude of having a linear utility where you only care about where you have where your utility just scales linearly with the outcome. But in general one would actually have a much more nuanced gradation or and tapering off of the utility and that is and if you want to capture all of that you one would not have to look at also the other moments of the other moments of the outcome. So, with this now let me I will just quickly end with by telling you the theorem under which there always exists such a utility function and that theorem is as follows what we will do is we will make we will make the following assumptions. So, we are going to make the following assumptions you note cap this fancy p the set of as the set of all lotteries it is the set of all lotteries equivalently the set of all possible probability distributions on outcomes. And if I gave you two probability distributions p 1 comma p 2 in p and suppose my set of outcomes let me denote this as o 1 o 2 dot dot dot o n. So, if I have n so that I have n possible outcomes and I gave you two probability this two lotteries p 1 and p 2 when I if I have by alpha p 1 plus 1 minus alpha p 2 what I mean is the lottery that gives probability under this. So, this is now a new lottery new lottery. So, this is also a this is also a probability distribution on O and if you look at any if you want to ask what is the probability that it assigns to any outcomes say o j the probability that it assigns is going to be alpha times p 1 of o j plus 1 minus alpha times p 2 of o j. So, this is basically and this is true for all j from 1 to n. So, in other words in other words when if you take a convex combination of of two lotteries here alpha is in 0 comma 1. So, when you take a mixture of these two lotteries. So, you have alpha times p 1 plus 1 minus alpha times p 2 what you get is a new lottery in which the probabilities are added up in the proportion alpha and 1 minus alpha. So, you take a weighted sum of the probabilities that you would get from the erstwhile lotteries p 1 and p 2. So, you can also interpret alpha times p 1 plus 1 minus alpha times p 2 as a lottery on the set of lotteries it is as if you are you are tossing another coin that gives you with probability alpha it gives you a lottery p 1 and probability 1 minus alpha it gives you a lottery p 2. So, you that is another way of interpreting this. So, with this we can now state our assumptions. So, assumption. So, the first assumption is the following there assumption is that there exists a complete transitive relation on the set of lotteries. So, which means that means for any p 1 p 2 in this set of lotteries we have p 1 less than equal to p 2 in under this preference relation or p 2 less than equal to p 1. We will also say that p 1 tilde p 2 or p 1 is equivalent to p 2 if both of these is true if p 1 is less than equal to p 2 and p 2 is less than equal to p 1. And we will use the notation that p 1 is strictly less than p 2 if p 1 is less than equal to p 2 but not p 1 is not tilde p 2. So, this is like a so one should think of this relation in the following way you the less than the this less this is like a less than equal to this is like an equal to and this is like a strict less than strict less than strict. So, the first assumption here assumption a 1 says that there exists such a relation there exists such a complete and transitive relation. Now, assumption a 2 says that if p 1 is equivalent to p 2 for all alpha in 0 1 and all p in in p alpha p 1 plus 1 minus alpha p 2 is equivalent to alpha p 2 plus 1 minus alpha p. So, what this is saying is that if p 1 and p 2 are equivalent then you can mix them in the same proportion with a third lottery p and that mixed lotteries would also be equivalent. So, the mixed lotteries here are alpha p 1 plus 1 minus alpha p and alpha p 2 plus 1 minus alpha p. So, they would also be equivalent. The third assumption is if p 1 is strictly less than p 2 then for all alpha in 0 1 and all p in in p you can. So, if p 1 is if p 2 is strictly preferred to p 1 then you can mix any other lottery with p with p 1 and similar and in the same proportion mix another mix it with with p 2 as well and the preference order would remain the same. And the fourth assumption is if p 1 is strictly less than p 2 is strictly less than p 3 then there exists an alpha in 0 and 1 such that alpha p 1 plus 1 minus alpha p 3 is equivalent to p 2. So, that means if you have three lotteries p 1 p 2 p 3 and there is a strict preference like this p 1 is p 3 is more preferred to p 2 p 2 is more preferred to p 1 then you can mix p 1 and p 3 in a suitable proportion alpha so that that becomes equivalent to equivalent to lottery p 2. So, what these four axioms you can see are very reasonable axioms they are basically saying that you have some way of of ordering ordering lotteries such that they there is such that the order is complete such that you know there is a certain this kind of mixing rules apply that if you mix p 1 with p and p 2 with p in a certain in certain ways then the order remains the same or if they are equivalent the equivalence is retained and if you have an ordering between three lotteries p 1 p 2 p 3 then also you can do some mixing and recover the intermediate lottery. Now, the theorem then is simple says simply this under assumptions under assumptions a 1 to a 4 there exists a real valued function u that maps outcomes to the reals called the utility function such that for all lotteries p 1 p 2 in the set of lotteries p 1 less than equal to p 2 is equivalent to the expected utility of the outcome expected utility of the outcome under p 1 is less than equal to the expected utility of the outcome under p 2. Remember the outcome itself depends the probability with which various outcomes arise depend on on p 1 it depends on the distribution and we are on the left hand side I am I am taking the distribution to be p 1 right hand side I am taking the distribution to be p 2. So, what this theorem is saying is that if these assumptions hold then there exists a utility function such that max such that comparing the expected value of that utility function expected value of that utility function is equivalent to comparing the two lotteries. So, in other words if your set of preferences or if your set of lotteries has satisfies this set of axioms then you are effectively what you are effectively doing is maximizing the expected utility with whether you like it or not there is implicitly underlying that a utility function which you are which you are maximizing. This is a landmark theorem because it really simplifies a lot of complicated matters like comparing lotteries and so on to a simple optimization question the one of maximizing the expected utility. It also gives you a lot of personalization because it tells you how different utilities the shape and the form of different utilities are really captured eventually by this by the preference with underlying preference relation and not just by the probability distribution the not just the set of outcomes. So, this is a this therefore is going to be a cornerstone of our form problem formulations in this in this course. I look forward to look forward to teaching you more about this. Thank you.