 Hello everyone, so in the previous portion what we had seen was that we had compared two different lotteries and we applied a particular logic which was to compare the cost of participation in the lottery with the average outcome of the lottery. And what we saw was we get a kind of unintuitive answer that by applying the same logic on two lotteries that are and the two lotteries are scaled versions of each other. We somehow get to this conclusion that we should be participating in both. However, intuitively it does not seem right. There is one lottery that seems more somehow riskier than the other and therefore our logic is somehow seems to not capture all the elements that are involved in coming to a decision about participation in this lottery. So just to recap what it was if you see here we had the two lotteries, the first lottery involved a choice involved participation. We wanted to participate in the lottery by we had to participate in the lottery by paying 100 rupees is also the worth of a cake. A cake costs us 100 rupees. In the outcome of the lottery you get three cakes with probability two thirds and zero cakes with probability one third. So the average outcome the average outcome of the lottery was that you get two cakes on average. So and now two cakes is worth more than 100 rupees because a cake is worth 100 rupees. So we said that it makes sense to participate in a lottery like this. But then I gave you another another lottery in which one had to pay not one 100 rupees but one crore to enter this lottery. Now one crore buys you a house of a certain area and the outcome of the lottery was that with two thirds probability you would get a house with price the area or and with one third probability you would get you would get nothing at all. In other words you would lose even the one crore that you paid for paid in participation to participate in the lottery. Now but once again the average outcome here was this was once again a house that is at least that is twice the that has twice the area. So it is three times three x the area here and with probability two thirds and one and zero with probability one third. So the average outcome was a house with so the average outcome was a house with twice the area and that was worth more than one crore. And so once again it is made it is the logic told us that one should actually be participating in this lottery. That one should actually be paying one crore to take a bet in which to you get with twice the size of a house with two thirds probability and with one third probability you will lose even the one crore that you paid for it. So this somehow seemed wrong because it seemed like we have not factoring in the fact that one crore means a lot more than the loss of one crore means a lot more than the loss of 100 rupees. And therefore there is an element of risk in the second lottery which is larger far far larger than it is in the first lottery. In the first lottery you would lose in the worst case just 100 rupees. In the second in the worst case you will lose one crore rupees. So there is a vast difference between the two and that has somehow not been factored in into the analysis. So looking the point here is therefore that looking at just the average outcomes the or looking at the point here is that looking at just the average outcomes the average outcomes do not adequately capture our attitude decision making under uncertainty. Decision making under uncertainty involves a lot more nuance thinking than just simply comparing averages and what this has this example has shown is basically that averages are not enough. So this taking this lesson forward now what we want to do is come up with we want to ask then what should one consider should one consider the average should one consider the second moment should one consider the third moment etc. So this is what the this is basically the question that we that we end up with. Now in order to so basically what we need in order to answer this sort of a question is a proper unified logical framework for comparing two different lot trees. In other and in when I say lot trees essentially what we need is a way of comparing two different probability distributions which is somehow better than the other. So what should be the framework for comparing the two is what we would be we would now be exploring. So in order to do this let us set up set up therefore a formal model. So formal model for decision making under uncertainty the model for decision making under uncertainty consists of the following. So you have first a set D which is a set of decision alternatives. So we want to we want to choose one out of this set of alternatives. This is our this is the set of alternatives that we have. The second element second thing is omega omega is called the set of states of the world or the states of nature. Now omega is a set of possible choices that nature is going to choose for us. We do not have any choice we do not have any control over what omega will arise. So omega is the space of a variable which is exogenous to the problem it is chosen by nature. So it is what so omega is what captures the uncertainty in this problem. Now every time we take a decision in a state of the world and the true state of the world is omega. So the decision is D here D in capital D and the state of the world is omega in capital omega. What we get is an outcome an outcome which is f of D comma omega. This is our outcome decision D and state and what we have also is a preference relation on the outcomes. So we have this what is this this is basically a relation which compares two outcomes. So this is a preference so it tells you preference relation between two on the set of outcomes. Let us say the set of outcomes is and here I will denote the set of outcomes is denoted capital O. So this less than equal to sign is a preference relation on the set of outcomes. Now what does this mean? It basically tells you if I give you two different two different outcomes what is the which is more preferred than the other. So when I say when I write outcome O1 is less than equal to outcome O2 it means that O2 is preferred that is all it means. So one way in which this can manifest for example is that there is actually a function. A function that maps the set of outcomes to say two real numbers and we have this and we have this we have a way of measuring the quality or the attractiveness of an outcome. So for example if G of O1 is less than equal to G of O2 then that is that we can say is equivalent to O1 being least less preferred compared to O2. So what one wants to do is find a decision that helps us sort of deal with the uncertainty involved. So now because the preference so you want to try and get the outcome in some sense of highest preference. But then the preference the outcome that we will get realized is not just a function of what you choose but also a function of what nature chooses because outcome comes about as a result of because the outcome comes about as a result of this function. This function is the one that it is a function of both the decision D as well as the state state of nature which is chosen by which is chosen exogenously. So therefore we need to have sort of some kind of a framework to say what is the way forward by which we are going to say a decision D is a good decision. So this gives rise to many so obviously there is no unique way of disambiguating this. So there are obviously several approaches. Some approaches for example are as follows. Here are some approaches. So one is to look at the average outcome. So one way is to look at the average outcome. So in this case what one has is actually not just an uncertainty but also a probability distribution over the set of possible omegas that can arise. So what one looks at then is you look at G evaluated not at any one particular outcome but actually at the average outcome. You look at G of therefore the expected outcome that will arise when you take a decision D. So the expectation here is with respect to omega where omega is the is that is now a random variable it is its value is chosen by nature and its distribution is also decided by nature. So you are taking an expectation with respect to omega and of the outcome that would arise when you take a decision D. So what we are doing here is we are proposing that suppose I took a decision D think of all the possible outcomes that can arise look at the average of them look at the value that that average outcome gives us under G and that is what we consider is the value of our decision D. So what we then say is well this is now a function of D. So what we can do is well choose the best D amongst all best D amongst all the possible decisions. In other words what one does is then you try to maximize this function over by choosing the best D in the set of possible decisions D. So this was basically the attitude that we use to compare the two in both of the lotteries that we just discussed essentially we looked at the average outcome. Another attitude is to look at the worst case. In this case one does not need to know the any particular distribution there is no probability distribution on the uncertainty we one just looks at the worst possible thing that could happen and then tries to take the decision that will give us the best outcome in the worst case. So you look at you look at so assuming you have taken a decision D and nature chooses a state of the world omega we think of all the worst possible state of the world that nature could choose for us. So you look at worst in the sense of the value in terms of the value of the outcome g outcome that will arise. So you look at the minimum of this overall omega and then we say let us let us see this is now the worst case divided or value that we can get out of a decision D. So considering that we have taken a decision D whatever nature could do cannot be worse than this and then we say well let me try to do now choose the decision that would maximize this worst case value. So you want to choose a D that would maximize this. This would imply this sort of attitude basically does not require us to know with what probability various out various states of nature are going to occur one just simply looks at the worst possible thing that could happen and takes a decision based on that. So obviously this kind of a this kind of an attitude could lead to many misses in the sense that you would miss out on opportunities that would have otherwise that you that you have ignored just because you are worried about a certain worst case. So clearly one can easily imagine scenarios where this kind of an attitude does not actually this is not actually work very well. The other extreme of this is to look at the best case this is to be perpetually optimistic you know permanently optimistic about about what will play out in the uncertainty. So you look at you think of the best possible thing that could happen when you take a decision D and nature chooses an omega and based on this you say well now let me choose the decision D that gives me the best possible thing in the best possible case. So you try to maximize this now over D in D. So these are these are some approaches that one can adopt towards decision making under uncertainty. Now it turns out that actually none of these none of these are actually complete in the sense that the kind of fallacy that we saw in the earlier in the earlier example will also arise with any of these approaches. So that there is a certain so what there is a certain better approach that exists and that is what I will talk to you about. So the approach that I am going to talk to you about comes from a field what that is known as expected utility theory. So this field is called the expected theory and this theory by in the course of its development automatically gives us a notion of risk. We will soon you will see as I go about developing it that risk comes out as a the risk and our attitudes towards various types of various types of lotteries that we had just discussed all of that actually comes out as a nice corollary out of this approach. Now what we will posit in this approach is that there is in fact a probability distribution on the states of nature. So states of nature occur with a certain probability. So there is a probability distribution set of states of nature or states of the world which is omega. Now look what happens here. So each decision that we take each decision can lead to multiple outcomes can lead to multiple outcomes based on the state of the world that gets realized based on omega. So you take a decision D based on the value of this is the decision that you took here is your decision based on the value of omega that means based on what nature does right state of the world based on what the state of the world is you could get various you could get various types of outcomes as a result of your as a result of the decision D that you have taken. Now you could also have the same outcome arising from various states of the world. So many different states of the world could sort of come conspire in such a way that you could still regard that they still for the same decision D give you the same possible outcome. So what you can do is you can say well since you are only concerned about outcomes we can talk of what is the probability with which you are going to get a particular outcome. What is the probability with which you are going to get a particular outcome when you take a particular when you take a decision D. This then gives you a probability distribution on the on all the possible outcomes that can arise. So it gives you a probability distribution on the set of outcomes. So this probability distribution is a function of the decision that you have chosen. Right? This because this the probability will change with the decision. Certain decisions will give you certain outcomes with higher probability and certain decisions will give you certain other outcomes with lower probability. So the probability distribution itself will depend on this distribution is what we denote as PD. PD of a certain outcome O is simply the probability. So let this probability distribution on the set of states of nature let this be denoted by P. So PD of an outcome is the probability of this set. It is all the it is the set of all omegas such that under decision D the omega somehow gives you the outcome that you are interested in this outcome given that you have taken a decision D. Right? So this this this now is a PD is now a probability distribution on the set of outcomes. And so if we have a framework for comparing the probability distributions on the set of outcomes we have a framework therefore for comparing decisions. Because we can then choose the probability distribution we can then choose the decision that gives us the best possible probability distribution on the outcomes. Right? So this this kind of an a lifting where we take where we go from decisions to outcomes and then from outcomes to probabilities on outcomes. This kind of a lifting is something you will see in other portions of the course as well. Because this is a way of thinking about problems under uncertainty that is extremely powerful. It works not just in these sort of problems but also in certain decentralized problems. You will see much more of this later in the course. So the benefit of this will become clearer you know more in hindsight for the moment just just bear with me that this what this has done is essentially it essentially it has given us a way it has given us a framework for comparing decisions provided we have a framework for comparing probability distributions on outcomes. Okay? So if we can compare if we can compare probability distributions on outcomes we can compare decisions we can compare and what is what is really a probability distribution and on an out on the set of outcomes well it is nothing but what we have been so far calling a lottery. So probability distribution on outcomes a simple word name for this is actually nothing but this is actually what we have been so far calling a lottery. In the lotteries that we had so far we had and there were two possible outcomes the outcome you know with either that you get thrice the three cakes or you get no cake at all. So those were the two possible outcomes and the probability distribution was 2 third and 1 third. Okay? So now so the fundamental premise of utility theory or expected utility theory is that one has a way of comparing probabilities or comparing probability distributions on outcomes and what is this way of comparing probability distributions on outcomes? So I will explain I will explain that to you expected utility theory the premise of expected utility theory expected utility theory presumes or imposes that there exists a function a function u. Now what is this function u? This function is a function from the set of outcomes to the to the real numbers. So it is a function u and this has this it has the following property that you would prefer you prefer decision 1 to decision 2 if and only if you prefer probability distribution induced by or the lottery induced by decision 1 to the lottery induced by decision 2 which is equivalent which is if and only if you this is the most important thing. You look at the expected utility the expected utility of the outcome under decision d 1. Okay? Look at the expected utility of the outcome under decision d 1 that is less than equal to the expected utility of the outcome under decision 2. So if so the term here this here is this term is the expected utility under decision d 1. So if I took the decision d 1 nature will choose it is omega I would get an outcome f of d 1, omega okay? I get an outcome f of d 1, omega the this is the outcome that I would get the utility that I derived from that outcome is u of f of d 1, omega. So the term here on the left is the expected utility. So the expected utility from this particular outcome from form this particular decision. So on the term similarly the term on the right is the expected utility under decision 2. So what the what expected utility theory presumes is that you would you prefer decision d 1 d 2 to d 1 provided the expected utility of decision d 2 is larger than the expected utility of decision d 1. So this what this has done is basically taken their set of decisions which were ordered in some in some sort of in some kind of a non-specific way because of because all you had is some way of some kind of preference ordering between them saying that you prefer this to that and so on and converted that preference ordering to a comparison of real numbers. You just compare the expected utility from each of decisions with each other those are just real numbers they are to be compared with each other and you choose the decision that maximizes the expected utility. This reduction is extremely powerful. Now one thing you will you would have immediately noticed that I said that I have used the word that expected utility theory presumes that there exists such a function. It is just presuming that there is such a function there is such a function u. It is only saying that it is not actually showing that there or proving that there exists such a function u. But does it what you will be what you will soon see is in fact it does prove such that there exists such a function. So what it will tell you is that if certain axioms are true about the space of about this particular preference relation. This preference relation which defines a preference between various between across different lot rates. So if you had if a certain set of axioms about the preference relation whole then there always exists such a function u. So although I have said here that the expected utility theory presumes that there exists such a function this is not the full story. Actually expected utility theory we will soon see that under a set of certain set of axioms about the preference relation between that compare any two lot rates on the set of lot rates on the outcome there is actually there always exists such a utility function. And it is the shape and the form of this utility function that will tell you that encodes in it the attitude towards risk. So all of this in the next portion.