 Okay. Yes. Thank you for the mind in me. Okay. Professor, we have seen the case for two lotteries or three lotteries. But what if there are more number of lotteries, like any arbitrary in number of lotteries, then you can generalize what I've said in a straightforward manner. So it's always a combined lotteries, always a linear combination of the simple lotteries. And it always results in a simple lottery where that will lie inside the polygon that is identified by the simple lotteries. Thank you. Yeah, so Professor. Yeah. So we could express the same thing by an M times N matrix, right? Yes, so the generalization I mean. So as you have seen, there is this P and K, which is essentially a matrix. And yes, so you can write it in the matrix form. And can this sort of, yeah, can this sort of algebra way of looking at it be useful in a particular way that's essentially this is essentially all linear algebra. Yes, yes. Okay. So can I go ahead? So let me look in the chat if there are questions. Okay, so. All the fine lotteries have same number of outcomes in general. Yeah, so all the lotteries are defined on the same space of all the lotteries are defined on the same space of outcomes. It's always defined on this set C. All the lotteries have the same outcomes. Of course, some of the lot of these lotteries that I'm combining can have zero probability for some of the outcome. So in general, each of the lotteries that I combine will have a subset of possible outcome that eventually includes the full set C of all outcomes. So do combine lotteries share some properties with Dirichlet processes, but I think they are much simpler than Dirichlet processes. But yeah, I mean, it's a very simple notion. NLK depend on more than one probability other than alpha k. Yes, of course, say now that we have find out that we can combine lotteries and get another lottery. We can combine combined lotteries and get another lottery. So any combination of lotteries will give you a lottery. So you can define as complicated. So this combination is an operation that you define on the space of lottery. And this will be very important for what we are going to discuss now. So let me resume and go ahead with defining preferences. Okay, so the way in which we define preferences is based on axioms axioms are simple requirement that look like say natural requirements. So the first one is that we are going to require that whatever preference relation we define on combined lotteries is going to be the same on the set on the corresponding reduced lotteries so that it is enough to define the preference relation on simple lotteries. So this also means that we are assuming that an agent with these preferences will be indifferent between a combined lottery and the corresponding reduced lottery. Okay. So this is looks like a very innocent assumption, but actually it is less innocent than it looks. Okay. But it looks like a natural requirement. Okay, second thing, second axiom that we are going to assume is the axiom of continuity. So continuity, I mean the best way to give you an intuition of what continuity means without entering too much mathematical detail is imagine that you have three lotteries and that lottery L1 is at least as good as L2 and at least as good as L3. Sorry, this should be in the reverse order. Okay. So this lottery L1 is at least as good as L2. Oh no, sorry. So this is in the right order. Okay, and L3 is at least as good as L1 and then two is at least as good as L3. There is a so continuity means that there will be a combined lottery between L1 and L2 such that L is equivalent to L3 in the sense that L, so the agent with these preferences would be indifferent between L and L3. Okay. So this implies that say the this set of lotteries between L and L2 are the lotteries which are at least sorry, between L1 and L are the set of lotteries which are at least as good as L3. And those between L and L2 are those for which L3 is at least as good as all this. Okay. And essentially these sets are closed sets. Okay, so this is continuity. Continuity is looks like also an innocent assumption but it is a lot of very strong consequence. One of them is that essentially preferences over lotteries are satisfied this axiom of continuity, then you can represent these preferences by a utility function. Okay, which means that for any two lotteries, if one is at least as good as the other, then the utility of the first should be at least as large as utility of the other one. Okay, so so this you have to think a little bit about it, but it's a really straightforward and direct consequence of the axiom of continuity. So the third axiom that you are going to impose is the independence axiom. And these also looks like a reasonable assumption to make. So we say that preference relation over lotteries satisfies independent axiom if however you define these three lotteries. If lottery L1 is at least as good as L2, then when you combine lottery L1 with L3, the combination will be at least as good as the same combination with the same weight when you take lottery L2. Okay. So, and this should be true for any alpha that you choose between zero and one. Now this is really very strong consequences and as a matter of fact, if you take these three axioms, then what the consequence of this is that the preference relation must be, must have a representation in terms of a utility function that has the expected utility form. So this means that there must be numbers un such that the expected value of every lottery is just the expected value of this you under the probability of the lottery. And what is this you when this you is is essentially the utility of a lottery that gives you outcome and with probability one. Okay. So why is this so. Well, this is so because the independence axiom essentially tells you that this lottery. So if you prefer lottery L1 to L2, then you should prefer this lottery here, which is the combination of L1 with L3 to this lottery here, which is the combination of L2 with L3. Okay. And now the same notion can also be extended to indifference. So if you are in difference between L1 and L2, then you should be indifferent between these two combined lotteries. Okay. And so what distance you is that what distance you is that the indifference lines, the lines over which you are indifferent between lotteries should be straight line and should all be parallel straight lines. Okay. And this is precisely the type of indifference curves that you have with unexpected utility because just simply because it is linear. Okay. So, so it is clear that an expected utility will have indifference curves that are parallel straight lines. But the converse is also true. I mean if you, if you have a preference relation over lotteries that satisfies independence axiom, then the indifference curves must be straight line parallel straight line, which means that there must be utility function that corresponds that represents this preference Now this also tells you how you, you can experimentally measure utility functions. This is numbers UN by making experiments. Okay. And what this is discussed in the lecture, but essentially you do different experiments until you find utility functions, lotteries over which the individual is indifferent. And then you know that the utility function that the reference relation should be straight line that the indifference curve should be straight lines passing through these two points. And, okay, so it's again a good time to stop for questions. And so, is everything clear. So could you show like two slides, I mean slides again. Yes, I can show again the slides. So, this one. One more. Next one. This result here is called the expected utility theorem. Yeah, so I had the question on this only I mean, does it like imply that you will have this kind of expected utility form, like if all three axioms I mean is it necessary. Yes, it's necessarily insufficient. So, the, if you have these three axiom, you have these three axiom, even only if the preference can be represented as I expected utility for, you know, expected utility for. Okay, and this expected to this expected utility form says that the indifference curves will be straight lines. Yes, will be straight lines with this. This you can easily check now because if you take two different lotteries. And you make say, if you make if you take two different lotteries. Imagine that you take lottery L1 and L2, and you are indifferent between them. Then you can combine these two lotteries into another lottery, which will be on this straight line. Okay. And because this is a combined lottery, then you will be indifferent between L1 L2 and this new combined lottery. Okay. So, other questions. So, no other questions that have been clear. Okay, so this is a very important result. So, okay, so let's, let's go ahead. Okay, so now the question is, these expected utility really describes how real people behave. Well, there's been a number of paradoxes that have been discussed that show that in particular situations. Do not really behave in the way that the expected utility would recommend. Okay, so, so take this example for example, so here is a situation where you are asked whether you prefer lottery L1 to L1 prime. So, L1 is the lottery that gives you C2 with probability one. Let's say that C2 is half a million dollars. Okay, so lottery lottery L1 gives you half a million dollars, whereas lottery L1 prime gives you with probability. So, 99% gives you two million and a half. With probability, with 10% probability gives you half a million, but with probability 1% gives you zero. Okay. And typically, if you, well, if you ask to people, which of these two lot that they prefer. They prefer to get the half a million for sure. Okay, because in lottery L prime, L1 prime, there is a small risk 1% that you will get nothing. Okay. Now think about a second question. Okay, you are asked whether you want prefer you prefer lottery L2, which gives you zero probability of C1 with probability 11% to get half a million and 89% to get zero. See whether this is prefer to lottery L2 prime, where instead you get with probability with 10% probability you get two and a half millions. And we will probably 90% you get zero. Okay. So, really, what they, what people will choose between these two lotteries is probably lottery L2 prime, because, well, because you have a 10% probability of winning two and a half millions. You have to 11% probability of winning just half a million. Okay. However, this behavior here is inconsistent with this behavior here, according to the expected utility. Why? Because essentially, if if you prefer lottery L1 to L1 prime, then you should prefer L2 to L2 prime. Why is this so? Because if you do the math, you figure out that the points, the lines that join these two curve these two lotteries L1 and L1 prime and L2 and L2 prime. They are parallel lines. Okay. So whatever is the expected utility that you are using, the indifference curves will always be parallel straight lines. Okay. So however you draw these parallel straight lines, then you should always have a preference relation where if L1 is preferred to L1 prime then L2 should be preferred to L2 prime. Okay. I hope this is, this is clear. If you want, I can just try to make. The more indifference curve, for example, is in this direction, parallel lines in these directions, then and the increase say downward, then it is clear that if you prefer L1 to L1 prime, then you should also prefer, should prefer L2 to L2 prime. This is because the utility and no matter how you change your utility, how you change these straight lines, whatever the straight lines are, you will always either choose L1 and L2 or L1 prime and L2 prime. Okay. So this is just one of the paradoxes of say choice under uncertainty. There are other paradoxes that also tells you that somehow one can distinguish between what is called risk and what is called, say, true uncertainty. So risk is when you know what are the probabilities, uncertainty is when you don't know what are the probabilities. Okay, so there are paradoxes that tell you that when you have uncertainty on the probabilities, then this expected utility, even though you can define a probability distribution over the uncertainty. But still, the expected utility paradigm runs into problems. Okay. I'm not going to into all these details. These are discussed in the book of Moscow L. I hope you can find a copy in your library. This is given in the reference in the website. But if you look at chapter six in Moscow and all these questions are discussed. So questions. So in the previous slide. Yeah. Like, can you explain why it has to be parallel in previously. Why do I mean, did you slide up to this. Elias paradox. Okay. Yeah, here, like, because you've chosen 1111 time to print in a combination that gives us, I mean, in what is this fire the parallel. Yes, the two green lines are part of them. I mean, is it because of the choice you made. Yes, it is because of the choice I made. And then why do you say that you can see there are really extreme choice because one of the pairs is close to this corner here and one of the pairs is close to this corner here. Okay. So, like, then you say in difference curves are perpendicular to this. No, no, they are not perpendicular. What I'm saying in difference curves are straight lines. They are parallel straight lines. So whatever they are, whatever is the orientation of in difference curves. Consistency requires that if you choose at one, then you choose should choose at two. Excuse me. You said we can find a similar examples in chapter six. I wasn't sure what was the source for that which book you mean by that. It's the book of muscle and Winston and the author, which is mentioned in the website. Yes, thank you. Excuse me, Professor. Yes. So for each parallel line we have a sort of orientation and if we go in that direction, the utility increases. So, no, okay, so the, so let me try to make. So, if you. So let's say, if you have a particular say particular utility function here would be something probably like this know because this is what you prefer the least. This is what you prefer the most so maybe the indifference curve of the utility would be like parallel lines in this direction. Okay, like this. Okay. The utility essentially increasing in this direction here. Okay, in this direction. Can you see the figure. Yeah, it's clear. Okay. So, but so the, what the expected utility theorem tells you all these three action turns you is that the utility function is that the preference defined by the utility function always corresponds to utility lines or if you want indifference curves that are parallel straight lines. Okay. Okay, perfect. No, no is clear. Okay. Okay. And again, it is say it is easy to, if you want, if you have a say, an individual that is making choice over lotteries with unexpected utility. It is easy to figure out what these two, what these lines are essentially because essentially you just need to find two lotteries over which the individual is indifferent. And then you have to find in which direction, in which perpendicular direction, the utility increases. Okay. So indeed, so this expected utility is useful and these axioms are useful formalization of how you would like to design a system that takes rational decisions. Okay. So, because, of course, when you would like that if this individual takes rational decisions then you would like these three axioms to be satisfied. You would not like say for example that your preference relation as discontinuities in the sense that you say prefer something to something else but then if you make something one of the optional little bit more risky than your preference changes completely. And so you want continuity, you want these independence axiom and also you want to say combined lotteries have a preference relation which is defined through the preference relational simple lotteries. Okay. Okay, so if there are no other questions. Then, let me proceed. So. Okay, so the last thing I want to discuss today is how do you define risk. Okay. Yeah, conclusion of the last paradox. I mean, is there any conclusion. I mean, any conclusion. The paradox you mentioned. Yeah, so the conclusion is that if you want to design a system that takes decision over uncertain outcomes in a rational manner, then the expected utility is the way to go. But that real people that we have. I mean, it may not be expected utility may not be a good model for how real people behave, because there are some special circumstances where you can show that it does not give reasonable outcomes. Like the alias paradox. Is this clear. But like the alias alias paradox came because we choose the individual to be rational right it's not because the individual is real. Yes. Yes. So the alias paradox tells you that the way which are rational. What would make decision in that situation is seems paradoxical from our psychological point of view. Okay. It means that the expected utility in some situation does not really describes what how we take decisions under and there is a whole field of behavioral economics that is studying what are I mean what are the systematic deviations of the way in which people take decisions with respect to the expected utility part of life. Okay. Okay, thank you. Okay, so let's discuss the risk and in particular this conversion so now one important point is that when, when we discussed preferences over simple outcomes without risk. What we said is that you can represent preferences with an utility function, but essentially that any monotonic transformation of the utility function would correspond to the same preference relation. This is not so in when you have a risky. When you have a risk involved. And why this is not so is that essentially the shape of the expected utility will tell us what is the propensity what is the attitude towards risk of the individual with those preferences. Okay. Let's see how this works. So. So now, let me consider that the set of possible outcome is actually the real line. Okay. And so you can think that the possible outcomes are monetary outcomes are amount of money. You can think as if we are discussing lotteries like horse lotteries. Okay. Normally, lotteries are equivalent to real random variables. And then the expected utility is nothing but the expectation value of the utility computed over this random variable that is just the integral over the probability distribution of the utility of the utility function. Okay. So, one interesting one useful thing that you can do is to define what is a lottery. So let me show a lottery that gives you with probability one. This, the expected value of the utility function. Okay, so let me put it more clearly so this L bar is a lottery such that the utility of this lottery is equal to the expected value of the utility of the law, the true lottery. So you can for every lottery, you can define this lottery L bar, which is essentially the equivalent certain lottery. Okay. So now, if this lottery L bar is preferred to the lottery L, then you say that the individual is risk averse. Why is this because essentially this lottery L bar is a lottery that does not involve risk. You just get L bar with probability one. Instead, the lottery L gives you as an outcome, a random variable X that can take different values with different probabilities. Okay. So, the difference between these two lotteries is that essentially the lottery L involves risk and the lottery L bar does not involve risk. Okay, so if you prefer the lottery that does not involve risk, then you are risk averse. Okay. And if instead you prefer the risky lottery, the random lottery to the lottery L bar, then you are called the risk lover. Okay. Now, it is easy to see that and this is essentially a consequence of Jensen's inequality that risk aversion occurs even only if the U of X is concave like in this in this plot here. Whereas your risk lover only, even only if the U of X is convex, so it has upward curvature. Okay. So let's see this a little bit more in detail. So, one way to see this is imagine that we have a lottery that can give all the outcome X one or X two with probability say one half, then the expected value of X will be this. Okay, you can compute what is the expected value. What is the expected utility. So the expected utility is the average of these two points here. Okay, with probability one half. With probability one half you have this point which is U of X one with probability one half you have this point which is U of X two and you end up with this point which is the expected utility. Okay. Now this is clearly less than the utility of the expected value of X, which correspond to this point here. Okay, if this is expected value of X, then this corresponds to this point. So you see that in this case and this would be the utility of the lottery L bar. Okay, because the L bar gives you as an outcome the expected value of X with probability one. Okay, so this is the case of a risk averse individual because the difference between the utility of the lottery L bar and the expected utility of the lottery L is positive. Okay, and you call this difference, the risk premium. So how much. So you call say this. You call this. Yes, this difference essentially is the risk premium. Okay. Also you can introduce what is called the certainty equivalent. Sorry. Yeah, so so you can introduce what is called the certainty equivalent to the sea of L. And this is the value of X, such that the utility of this value of X is equal to the expected value of you. And if you go to this graph, then you have to take this value of the expected value of you, and you compute this value of sea of L. Okay, so you see that when you are risk averse. This certainty equivalent is less than the expected value of X. So, so this difference is what you are willing to pay in order to get rid of the risk. Okay, and yes, this is called the risk premium. Okay, this is the expected value of X minus sea of L. And so of course, if you have instead utility function that has the opposite, the opposite curvature that it goes like this, then, then all these relations are inverse, in the sense that the risk premium becomes negative in the sense that you are willing to pay in order to play this lottery. And if your utility function instead is just a straight line, then the risk premium is exactly equal to zero. And, and this is when this utility functions are called the risk neutral. Okay. As you can see, there are a few questions on on the chapter. So, let's. Okay. Okay, so. Yes, so. So, not to be the science designers should design lot that is using expected utilities. Well, say, Well, this is, say, a rational way of designing lot Teresa. And the other can say, when you think really think about lot Teresa, there are also other considerations that because essentially here we are considering lot Teresa. Just monetary outcomes. No. So, if you think about horse races, for example, these are lot there is where you buy you, you, you, you buy a ticket. In order to to run this lottery. Okay. You can buy a ticket for a particular horse on a better of a particular course or horse. And now of course, you can buy one ticket to ticket three tickets so how much the ticket cost. Should take into account that the number of tickets that can be both or can be very large. And so these are slightly more complicated problem, and it's very similar to the problem in finance. In finance. What people have to do is to decide what is the right price of asset that can be. For example, an insurance contractor. But once you publish this price, then people can buy as much as they want of this asset. Okay. So, the way which we will discuss at the end of the course how you fix prices in that case but more or less. Any type of concept. Entering play. Okay, there was a discussion on Jensen's inequality and it has been clarified for then there is a question on the masterful book. Yes, which has been answered. So, very good. So, other questions. Hello, Professor, can you come to the slide. Yes, then. Yes. One more back. Yeah, this fourth point what does it like what is saying like what is physical fourth one yeah. So essentially this lottery L one L bar is the lottery. Such that the utility of this lottery is equal to the expected value of your own. So it's a lottery. Yes, it's a lottery that gives these outcome with probability one. Is this kind of thing or like what sorry with probability one means what like means it is sure event kind of right if you do this lottery is sure. Sure, yes. It's not a random lottery is is a, it's just a lottery that gives you something with probability one. Okay. Yeah, you can think of it as a lottery that is a probability distribution which is a delta function. Okay, thank you. Yes. Okay, so I think our time is over. So, thank you very much. So, and well, on the website, you will find a little bit more about this subject. You. I mean, there is a discussion of, if you think about utilities, utility functions. Utility function which have particular properties of risk aversion. And you can, given a certain utility function, you can ask yourself whether this describes a risk averse risk neutral and how this risk aversion changes with with amount x. Okay, but I think the main concepts are the one we describe now. Okay, so next. Next lecture will be next week. Yes. Yeah, may I ask you one question. Yes, please. Yes. Okay. Okay, so I think we will have the function like concave because, okay, according with dancing, but because the, the investor or the other person who is defined this utility will be received a risk will be benefit for for the risk free. Because of the risk free that this is expected to, to receive right, it's a premium risk, sorry, for the premium risk, right. Because this is recent bears, and this will be probably lose or probably get with uncertainly we, and this is a very high risk, right. Essentially, so if you are risk averse, you are willing to pay in order to get rid of the risk. This is what you do when you buy an insurance. Okay, so you're, I mean, you can get an accident with your car and you may lose the value of your car. So you go to the insurance company and you get an insurance. This means that if it doesn't happen, then, when you're, but if it happens, then the insurance company will pay for your car. For my car. Let's say, and so we are choosing to pay something which is not random, which is sure, which is the insurance premium over you about a lottery in this case a lottery which can have a negative outcomes for you. If I'm, I'm out, outcomes for me. Okay. That was in my risk premium is negative. No, the risk premium is positive because it refers to the concavity of the curve. Okay, the concavity of the course. Okay. And what is the relation with this normalization distribution in the concavity in the course to the line of the normalization graphic. Sorry. It's them. It is like this light on this reason they're dancing. Oh, sorry, sorry, sorry, sorry, sorry, let me get to the slides. So you mean this graph. Yes, because this is the normalization graph, and then this is the concave graphic that is the James inequality. Yes. The normalization graphic of distribution. This is the utility function. Yes. So the black line is meant to be the utility function. The yellow line is meant to be the probability density function. This can be anything. Anything. Okay. Okay. Thank you so much. Okay. Okay, so thank you very much. So let's take a 10 minutes break before joining. We don't a second lecture so I'll assign you to break out rooms in a random way. Okay. See you later. Okay. Very good. So people are joining back.