 Hello everyone, so towards the end of the previous lecture we saw one of the most important theorems in expected utility theory and that theorem basically gave us the reason for why we should be using the maximization of the expected utility as our way of making decisions. It basically what that theorem told us that if a certain set of axioms hold then finding certain set of axioms about our preference ordering on the set of lotteries hold then we really have no choice then there must exist a utility function which maximizing which is equivalent to ordering our lotteries. So in other words finding the best lottery and or equivalently the best decision is equivalent to maximizing the expected utility. So before I move on let me I will just tell you a few characteristics of the theorem one of the few points about the theorem. So let us take note of the axioms that we had. Our first axiom was that the set of lotteries has a complete and transitive relation given by this curve less than equal to sign. This is the which so what is a complete and transitive relation it is complete if every two lotteries can be ordered and it is transitive if you have the following simple intuitive property that if p1 is less than equal to is less is less preferred to p2 and p2 is less preferred to p3 then it must be that p1 is less preferred to p3. So this is effectively just a logical consistency condition that if you prefer p2 to p1 and if you prefer p3 to p2 then obviously you prefer p3 to p1 as well. So this was our first axiom. The second axiom is very reasonable it said that well if p1 and p2 are equivalent if you are equivalent to each other which means you are indifferent between p1 and p2 then if I gave you another lottery p and I gave you and I mixed p1 with that lottery p. What does mixing mean mixing if you recall what I told you was that it is you can think of it as if there is another that is you think of it as if there is you are doing a coin toss where with probability alpha you would be choosing p1 and probability 1 minus alpha you would be choosing p. This now is another lottery on the set of outcomes and this lottery is has this lottery is basically is denoted by the left hand side here. The left hand side alpha p1 plus 1 minus alpha p is essentially the probability distribution that you would get from a lottery like this. So what this what axiom 2 tells us is that this kind of mix this kind of lottery on top of lotteries or a mixture of lottery p1 with lottery p in a proportion alpha to 1 minus alpha will also be equivalent to mixing p2 with p in the same proportion alpha to 1 minus alpha. So in other words if I if with probability alpha if I with probability alpha if I replace this with p2 then for you if p1 and p2 are equivalent then the resulting mixed lottery would also be equivalent. Axiom A3 tells you the same version of tells you a version of this if p1 was strictly less preferred than p2. So if p1 is strictly less preferred than p2 then mixing p1 with p and mixing p2 with p in the same proportions are in the same proportions alpha is before. So taking alpha p1 plus 1 minus alpha p or alpha p2 and 1 minus alpha p should give you should should continue to maintain the order of preference. That means alpha p2 plus 1 minus alpha p should should still be more preferred than alpha p1 plus 1 minus alpha p. Now axiom A4 is a continuity axiom. Axiom A4 effectively says that well if you have 3 lotteries p1, p2, p3 and they are the preference order is in this way that p1 is less preferred to p2, p2 is less preferred to p3. In that case in that case you should be able to find a mixture of p1 and p2 sorry of p1 you should be able to find a mixture of p1 and p3 in such a way that by suitably tuning the mixture you should the resulting lottery the mixed lottery on p1 and p3 should be equivalent to p2. So you should be able to find an alpha in such a way that when you create this new lottery on new lottery on on p1 and p3 so with probability alpha you you get p1 with probability 1 minus alpha you get p3 this sort of new lottery should be equivalent to this should be equivalent to the lottery p2. So if so for every such p1, p2, p3 that are ordered in this way you should be able to find an alpha like this. So this is effectively saying that there are that that there is a continuum of of lotteries that that that are available to you in you can and you can you can find if there is an intermediately preferred lottery that is that can be realized using a lottery on top of lotteries. Right. So under this assumption what did what did we see we find that well there that the preference relation p1 less than equal to p2 is equivalent to the expected utility of the outcome being less than equal to the expected utility. Remember this is simply a preference relation that satisfies the axioms that that I have listed whereas this here is the numerical less than equal to. So what what what has happened is that you have a some set of preferences on on the set of lottery that are described through your axioms and what has what this theorem has given you is that there is a characterization of that preference relation in terms of a numerical optimization an optimization of the utility of the utility of the outcome where the expected utility of the outcome where the expectation is taken with respect to the lottery that or the probability distribution that you are considering. Right. So this this is effectively told us that so now we can move to lecture 4 and what this is effectively telling us that your preference between decision so if you you prefer d2 to d1 which we said was equivalent to saying that you prefer the lottery induced by d1 to the lottery induced by d2 and that we are we we now see is equivalent to the expected utility of the outcome under d1 being less than or equal to the expected utility of the outcome when you take decision d2. So in other words max in other words the problem of finding the best decision finding the best decision is becomes equivalent to maximize the expected utility of utility of the decision. One other so so this becomes a way of for us or this becomes a way for us to take find the best decision out of a set of decisions. One other thing I I did not mention about the theorem is that this utility function U is actually is actually unique essentially unique. It is essentially unique means that if there is another if if if u tilde is another so if u tilde is another utility function function satisfying satisfying satisfying this relation satisfying the relation star then it has to be that u is equal to some scalar s1 times u tilde plus a scalar s2 where s1 is greater than 0. So so what this is saying is that well you the utility function that that the theorem provides is actually unique up to a scaling and a shift. So you can scale it by by a constant s1 greater than 0 and you can shift it means you can add another constant s2 to it and that does not change that sort of a utility function will still satisfy these these problems. So so so these these two operations are actually something that since we are maximize since we are finding the best decision D by maximizing the expected utility these two operations do not change the maximizing decision. The the optimal D that comes out of this problem and the same that would come out of the the problem where this u is replaced by if I replace this u by u tilde if I replace this u instead by u tilde the optimizing D would remain still the same so long because u u and u tilde are related by this particular relation. So this is this is an this this is what the expected utility theory and the expected utility theorem basically teaches us. Now what is the what kind of utility what kind of a function is this utility function and what does it encapsulate what meaning does it encapsulate and what are the things that it implies. So let us let us look at this a little bit a little bit more. In in our previous lecture we had looked at a utility function in which we we had taken this particular utility function we where we took that the we had taken in the previous lecture we had taken this particular utility function if you if you recall we had taken the utility function where utility of an outcome o is alpha o minus o square right. So this this particular utility function if you see I had mentioned to you that this function is is a concave increasing function and as alpha becomes larger this function tends to become more and more linear. Now is this is this actually turns out to be a very generic property almost in most cases pretty much every case the utility functions have this this shape they are utility functions are typically are typically concave and increasing you and as we as we saw the quadratic utility function that we took alpha alpha times o minus minus o square was also of that of that kind of nature. So if they are concave and increasing then what this means is that because it is increasing u dash of x is greater than 0 usually and u double dash of x is less than 0 this is the this is usually the form of of a utility function. So this is of course assuming that the utility function is differentiable but under this as an under this assumption this is what this is this is this is how the utility function looks. Now this only tells us that it is concave and increasing does not tell you I fixed any particular form for the utility function. Now whether it would be quadratic whether it would be linear whether it would be whether it would be exponential or whatever really depends on the kind of preference relation that you have to begin with. So it really depends eventually on this this curved inequality the preference relation which satisfies all the axioms it encapsulates the it characterizes eventually the shape of the utility function. So based on the kind of a preference relation you would have your expected this problem of maximizing the expected utility can be quite non-linear eventually and in the most general case this would be a very this would be a a non-linear optimization problem in which you would be some some fairly nuanced or barely complex complicated concave increasing function. In so in that case therefore what what this effectively is telling us is that the the because because u is in general going to be non-linear the the expected utility when expressed in terms of the moment in terms of omega would in general have a contribution from all the moments all the moments of omega and not just the mean. So because this will have the contributions from all the moments of omega effectively it is saying that every moment would every moment matters every moment of omega matters. So so so this effect this also tells us the where we were going wrong when we were just looking at only the averages and and also answers the question of what actually should one look at well the we were going wrong in looking at the averages by basically ignoring all the higher moments. So if you if you one can think of it this way that you can think of u as as as basically as some kind of a power series suppose so it is it is say a 0 plus a 1 a 1 x plus a 2 x square plus plus a 3 x cube and so on and then and suppose suppose we have suppose suppose for simplicity suppose f you know f was suppose linear just like we had suppose f was linear then in that case you would you would get some you once you if you look at the expected utility maximization problem the expected utility of this would in general have would in would in general have contributions from contributions from say suppose some b 1 into into omega plus b 2 into omega expectation of omega square plus b 3 into expectation of omega cube and so on and of course additional terms in x and the and additional terms in x additional terms in D where these b 1 b 2 etc all of these would also be functions of D. But the main lesson here is that you would because because in a general nonlinear once the utility has a general nonlinear form you cannot wish away any particular moment you know the the moment various moments all the moments of omega could in general matter and our fallacy earlier was to somehow in per force in for you know enforce that only the first moment should it should be considered and all these later moments should it should for somehow be ignored. So, this is this is where we had gone wrong. Let us also now see how this this sort of utility function explains the cake versus house problem the dot v choice that we had or the contradiction that we had seen. So, let me draw a utility function like this. So, I am going to draw a utility function and what I want let us try to write out the expected utility and the cost of the expected utility from the lottery and the cost of entering the lottery. So, this is this is my this is my utility function what I have drawn here is is the outcome. So, this is 0 cakes this is 1 cake this is 2 cakes this is 3 cakes. Now, the cost the let us suppose somewhere here is is suppose here somewhere is 100 rupees. Now, the remember the expected utility that we that we get from the expected utility that we get from the lottery was two thirds the utility of of 3 cakes plus one third the utility from 0 cakes and that I am going to take as 0. So, it is going to be therefore two thirds the utility of of 3 cakes. So, if I look at the utility that I get from this is my utility function this the utility that I am getting from 3 cakes is out here. So, this is therefore the line here is utility from 3 cakes this is the utility from 3 cakes then in that case two thirds of that would be say somewhere around this height. So, two thirds the utility from 3 cakes is this height. So, let me write this here. So, maybe two thirds the utility from 3 cakes and as you can see that is in this case what I have shown is that what I have depicted is that is actually greater than that is actually greater than 100 which was the cost of entering the lottery and this is the expected utility from the lottery and this being larger than this implies that I would actually prefer entering the lottery. Now, let us now suppose I take this problem increase the stakes. So, when I increase the stakes this 100 is going to be this 100 is now going to be raised even high is going to be raised higher higher eventually to something like 1 crore. This is what it would cost me to participate in the lottery this would be very high somewhere here. The utility from 3 houses would be somewhere here. So, if you look at so I am going to just extend this just extend your imagination a bit and from where here maybe is the utility from is the utility from 3 houses. So, utility from 3 houses of course, it will be much more than the utility from 3 cakes but the point is that because this is a concave and increasing function effectively what is going to happen is that the increase is going to taper off. This is a function that increases but then eventually starts increasing at a slower and slower rate that is because it is concave that is u double dash is less than 0. So, because it starts increasing at a slower and slower rate it effectively means that they will come a time when the stakes become so high that the utility does not increase in proportionately. So, if I take a point somewhere here say for instance then this would be for instance my utility from a house of thrice the size two thirds of that. So, this here would be my utility from a house of thrice the size utility from 3 houses like this utility from a house of thrice the size and then two thirds of that would probably be somewhere here. So, this here is the two thirds times the utility of thrice the house. So, in the second lottery what is going to happen what is going to happen is that the expected utility from the lottery is going to be this red level here which is two thirds the utility of thrice the house but the cost of entering the lottery has gone way past that. The cost of entering the lottery is now here which is the green level the green level here. So, because now the cost of entering the lottery is higher than the utility that you would get from the lottery the expected utility from the lottery it automatically is telling us that that you would not want to prefer you would not want to enter this lottery. So, the earlier lottery was okay with you but the new lottery is not and because and that is because you have the stakes have risen so much that and the cost has increased linearly but the utility from it the expected utility has not grown linearly. The utility sort of tends to taper off and tends to grow slower and slower and as a result of that one does not one tends to prefer the one crore for to one tends to prefer keeping the one crore rather than get into a bet like this okay. So, this is what this is this is this comes out as a way of as a natural corollary of having developed the expected utility theorem. The expected utility theorem also gives us a very concrete and quantitative measure of risk now risk can be measured in many different ways but here is one particular measure so which I can which I will just tell you about. So, we say that a decision maker a decision maker is said to be risk averse decision maker is said to be risk averse if he has this if his expected utility with this you know computed under any probability distribution P is less than equal to its utility from the average out under that same probability distribution P. Now, this is a this inequality here is actually an example of what is called Jensen's inequality and this holds whenever U is concave. So, if U is concave then you would always have this property that the expected utility is always less than equal to the expected utility is less than equal to the utility of the expected outcome or the mean outcome. So, the expected utility on the left hand side is essentially the utility that you would get from the notary. The utility of the expected outcome is the utility that you would get when you when you are given with certainty the average outcome of the lottery. So, the right hand side the right hand side here is the utility the average outcome is given with certainty of the when the average outcome is with certainty and this is the utility of the lottery. Now, what does this mean for what does it mean for a decision maker to be risk averse it basically means that a decision maker does not prefer the lottery as compared to be given the average outcome of the lottery with certainty. So, although the lottery gives gives him a chance of getting more than the average he always prefers getting the average with certainty as compared to participation in a lottery in which the in which you know things could go either way. So, this is what it means for a decision maker to be risk averse. So, most decision makers essentially and most ways of posing a problem are problems have in them this property that they involve they assume they have utility functions that are risk averse. So, a utility function would be any concave utility function would have this particular property. However, there are some exceptions where there are some individuals who are risk loving or whose risk seeking or risk preferring in which for whom this inequality actually reverses. So, they prefer they enjoy the risk they prefer taking the risk as compared to taking the outcome which is which is available with certainty. So, we will not go and get into those kind of problems in this course. So, a measure of one measure of risk here is a measure of risk that can that we can talk of is as follows. Now, motivated from here we can ask the question. So, suppose if you had the option of if you had suppose you had the option of buying insurance to get get away from get away from uncertainty. So, suppose someone made you this offer suppose you pay y an amount y to get the average outcome outcome with certainty. So, then what you would have gotten is you would have got you have given you have you are receiving the average outcome with certainty but you are paying y for it. So, now question is what is this what should be the y for which what is the maximum y that you would be willing to pay. Well, it is the y for which the utility from the average outcome and now here I am assuming the outcomes are all just scalars. So, the utility from the average outcome minus the y that you have paid this this is what you are going to get with certainty right you are going to get an average outcome and you are also going to be you are also going to be poorer by an amount y. So, the utility from receiving the average outcome minus y that you have already paid for it this should be equal to the expected utility the expected utility of the outcome. So, here where x is a random variable. So, if you have a random variable x then you can ask what the the maximum y you can ask what should be the y I should pay to get away from the uncertainty in x well the amount that you the maximum amount you will be willing to pay is the y such that u of expectation of x minus y should be equal to the expectation of u should be equal to the expectation of u of x right. So, now we can try to do the following approximation. So, suppose I approximate with you know some kind of Taylor series type of approximation I approximate this as roughly u of you expected of the u of the average outcome plus sorry minus y times u dash of the average outcome plus terms that are very that are that are say small in y. I can also do the same for the other for the term on the right hand side I can write expected the expected utility as say the as again using using the Taylor series of expansion can write that this is the expectation of u of u evaluated at the average plus deviation of x from the average times u dash evaluated at the average plus the second d plus half times the x minus the average squared times u double dash evaluated at the average plus a bunch of other terms that are that that are small o of x minus the average of x there is the expectation of all of this. So, now equating these these two what we find is that so you can see a few things cancel out here. So, this this here when I take the expectation of this particular term this is going to be 0. So, the u dash evaluated at the average is going to disappear this term is a constant it will it will come out of the expectation. So, what we are eventually left with is is the an equation like this an equation where we say where we find that the the y that you need to pay is is equal to some is roughly equal to something like this it is negative half sigma square u double dash the average you double the you double dash of the average outcome divided by u dash of the average outcome plus plus terms that are where what is sigma square well sigma square is nothing but the variance sigma square here is the variance is the variance of x sigma square is the variance of x. So, this this coefficient this particular thing that we have here this quantity here actually the negative of that the negative of u double dash divided by u dash is r may is you can think of this as a measure of risk aversion. So, you can the r of x which is defined as negative of u double dash x divided by u dash x this is a what is called the aeroprat measure of risk aversion and it all what this measure basically is telling you is how much insurance would you be what is the maximum insurance you would be willing to pay to get away from uncertainty and get the average outcome of the lottery instead to get in exchange for participation in the lot in exchange for participating getting into the lottery itself how much insurance are you willing to pay to get the average outcome of the lottery and that that the amount you are willing to pay scales roughly as something like this be with this coefficient u double dash x divided by u dash x you can see also that this this this measure of risk aversion actually increases with increases with x sorry decreases with x. So, what this means the reason for that is usually so what this means is people effectively as their wealth increases as the x increases people become less and less risk covers. So, this all of these beautiful consequences are coming out as a result of result of a formal way of thinking about the utility or and about the expected about thinking about decision making under uncertainty in a formal way and through the expected utility theory. So, we will discuss more on the other side of in the next lecture.