 Hello everyone this course is called stochastic control and communication although these are the two words I have used in this course what I will try to persuade you towards the end of the course or through the course or through the course of the course is that there is that stochastic control communication several problems in collaborative decision making team decision theory like problems in economics, organization theory and so on all of these can be thought of in one common framework. They are actually all instances of a larger kind of theme of decision making under uncertainty and the way these problems, the way the decisions and the uncertainty and the information and the dynamics all of these things the way they interact with each other give rise to these various different problem classes that have been studied in as different you can say different disciplines across academic across all of academia. So, what I want to do through this course is to sort of persuade you that one can look at all of these problems in unison in under one kind of lens and under one kind of common way of thought. In that sense this is actually not a regular course in stochastic control or regular course in communication it is sort of a blend of the two or a way of trying to look at both together or look at both through a common lens. So, welcome to this course I hope you will all take back this view from this course view that is very close to my heart and is one there where on which I have worked on for several years I hope you will all benefit from this. So, as I said the idea is to think of the following the following topics one stochastic control second is information theory or communication theory team decision making collaborative decision making and several problems in economics several problems in economics all of these should we would like to think of in one common framework. So, one common framework that equals this course. So, you might wonder what is what is common to all of these these different ways of decision making. So, what is common to all of these different ways of different types of problem classes. So, what is common is that they all fall under this broad umbrella of what I call decision making under uncertainty. So, now decision making under uncertainty has two key aspects So, which means what does this mean there is there are two key aspects first is there is a decision a decision is some kind of variable whose value you want to choose it is and one of the and set of alternatives that you want to choose or something like that. So, this is a decision that you need to make and obviously then the other aspect is that there is uncertainty. Now, uncertainty what do you mean by uncertainty? Uncertainty means that there refers to a an aspect or a variable in the problem whose value is not fixed. So, variable whose variable whose value is not fixed or not constant it is also not known at the time the decision is to be made is not known is not let me say not fixed and not known at the time of making the decision. So, decision is a one choice out of a set of alternatives is a choice from a set of alternatives. So, this could be a value for a particular variable or one element from a certain set and uncertainty is a variable whose value is not fixed and not known at the time of making the decision. So, in the absence of uncertainty absence of uncertainty all elements of the problem are all elements of the problem are fully known and have definite values in the absence of uncertainty we can say the whatever is the variable that captures the uncertainty already has one fixed value and that fixed value is already is known to you when you are when you have to make this particular when you have to make your decision. So, as such you can say you can think of this as a kind of well posed decision making problem because or a very well or a very you can say in some sense an easy decision making problem because all you have to do is think of that one particular value for that for the for the uncertain variable and ask what is the best decision when this variable takes this particular value. The problem with uncertainty that arises with uncertainty is that the uncertain variable can take many different values and therefore, thereby lead to many different scenarios and as a result of that you need to you know that the kind of decision that you need to take may vary with the scenario, but you cannot choose a decision that varies with the scenario. You cannot choose a decision that is that is tuned to different scenarios or to different values of the variable. So, you have to choose a decision before even the variable makes its value known to you knowing only the only certain characteristics about the variable. Say for example, knowing only that the variable can take a certain range of values or knowing a probability distribution about the variable or something like that. So, in that sort of framework you need to take you need to take your decision. So, this is the issue this is decision making under uncertainty. So, and what I will persuade you through this course is stochastic control, information theory, team theory and several other problem classes all actually have this one common, common flavor to all of them that one has to make a decision with the lack of knowledge of a certain uncertain variable. So, these are the two key elements of a decision making problem under uncertainty. Now, whenever there is uncertainty involved in a problem this it results in two kinds of phenomena or two kinds of issues or aspects that one has to consider. So, two kinds of two important aspects decision making under uncertainty, two key aspects. First aspect is that there is an element of what we call risk. Decision making whenever there is uncertainty there is automatically there is automatically an aspect of risk. Now, I will tell you what this risk by risk you may think of this as a word from English with an English meaning that something is risky say a road is very slippery therefore, it is risky or an asset is extremely volatile therefore, you think it is risky or betting is risky or something like that, but there is a very precise way in which we can think of risk. So, I will talk to you about that. So, but the important thing to consider is the way we make decisions whenever there is uncertainty involved risk is endemic you cannot sort of wish away risk. So, risk is an almost an integral part of any decision making problem under uncertainty. Now, I can I question you by risk I do not mean the English meaning of risk there is a certain formal way of thinking about defining what I mean by risk and I will come to that. So, one is risk the other aspect is that of information gathering see when we are when we have to make decisions under uncertainty obviously, because you do not we are we are choosing a decision before the before we know the value of the uncertainty. So, consequently one of the things we would want to do is to know more and more about the problem more and more about the uncertain variable and this this aspect therefore manifests itself in the issue of information and information gathering information transfer information leakage and so on. So, the role of information is again a fundamental aspect to dissimilar to problems of decision making under uncertainty. The way in fact the nature of the problem the character of the problem the hardness of the problem all of this changes dramatically as if you change the way I give you or change the assumptions under which I give you the information. So, the way I the manner in which the information is revealed the sequence in which it is revealed the timing at which it is revealed all of this changes the way the nature of the problem and all of this gives rise to different types of problems. So, bear in mind that information gathering has a fundamental role also in addition to risk in problems of decision making under uncertainty. So, now let me let me go through these two issues one by one. So, the first issue is that of risk what do we mean by risk? So, if you want to understand what I mean what you mean by risk let us consider the following let us consider the following one particular example of what we mean of what risk could of what that sort of tells you what risk is. So, suppose you are suppose you have let us say suppose there is a cake and that cake is worth 100 rupees now I tell you I tell you that you have the following you have the following option you have the following lottery you pay 100 rupees you pay 100 rupees to enter into this lottery. So, there is a lottery now you pay 100 rupees 100 rupees to enter this lottery now and the lottery is as follows. So, if you now what I will do is I will toss a coin and the coin is not a fair coin with two thirds probability it will come it will give me one outcome and one third probability it will give me the other outcome. So, I pay 100 rupees to enter to enter the lottery I draw my I toss my coin to enter the lottery that a coin is tossed if so and with two thirds probability with two thirds probability with probability two thirds I will get a cake I will get three such cakes and with probability one third I will get zero cakes the question is would you be interested in this lottery the cake is worth 100 rupees it costs you 100 rupees to also enter the lottery with when once you enter the lottery that the your the lottery is as follows with two thirds probability you will get three cakes and with one third probability you will not get any. So, question to the question that is being asked is would you be interested in this lottery now there are many different ways in which one can approach this problem the simplest way is to say well what am I getting on average what am I getting on average well on average what I am getting is two with probability two thirds I am getting three cakes the probability two thirds into three and with probability one third I am getting zero. So, what I am getting on average is actually two cakes and with the 100 rupees that I had I could have bought I could have I could have potentially bought I could have potentially bought just one cake and I am therefore I compare the 100 rupees that I have which is which for me is worth one cake and on the other hand this lottery which on the lottery on average is giving me giving me two cakes. So, I would rather prefer I would prefer the lottery. So, the logic again let us let us recollect logic again. So, one kind of logic let me put this in quotes because I will later argue that this is actually not a complete logic. So, one possible logic. So, you have take the average outcome the average outcome is two third into three cakes plus one third into zero cakes that is equal to two cakes and the cost of entering the lottery is actually is rupees 100 and 100 in 100 or can buy you one cake. So, on rupees 100 can buy you one cake. So, as a result since the cost of the lottery is so it seems therefore that the cost of lottery is less than the average outcome of the lottery is I would prefer I prefer the lottery. This is one particular one one kind of one sort of logic. So, this what this logic what was the train of logic this logic it looked at the average outcome it looked at the cost of entering the lottery it compared the average outcome with the cost and said well if the cost of the lottery is less than the average outcome I should I should actually average outcome of the lottery it makes sense to be in the lottery than to not be in it. So, that is that is that is the logic here. Now, let me show you why this logic fails there is a there is a significant flaw that is that is that is sort of built into this logic. So, let me let me ask you the let me show you for example, if this was not if this was not 100 rupees for a cake but rather say for example, one crore for a house. So, the question then is you have it is not a cake anymore consider this other lottery consider this other lottery. So, what do you do in this other lottery you pay you this lottery it cost you rupees one crore to enter the lottery I mean not all of us have rupees one crore but let us just assume for suppose you have this one crore to enter the lottery. So, I am just exaggerating for the sake of to making a point. So, we suppose it cost you rupees one crore to enter this lottery and the lottery again makes you the same offer it will give you with two thirds probability it will give you a house that is this this. So, and this rupees one crore can buy you say pay rupees one crore to enter the lottery and one crore buys a house of a certain size of a certain area. So, and then offer that is made in the lottery is against the same it says with well probability two thirds with probability two thirds you will be you will be you will be given a house that is thrice the size three three x house with three x area and there is a one third probability that you will not get anything at all you will not get any house at all. So, question is would you bet rupees one crore to get into this lottery. So, if you if you if you bet if you bet one crore and it turns out that and the second outcome is what comes out then in the if you bet one crore and this outcome comes out great you get a house that is thrice the size if you but it can also happen that you do not get a thing you do not you lose your you lose your one crore and you do not get anything at all. Now, if I follow the same logic that you had outlined earlier that the logic that is there on top of the top of the page it is looking at the average outcome the average outcome here is still now in place of cakes you have a house. So, the average outcome is then a house that is twice the size the average outcome is a house that is now twice the size and the cost of entering the lottery is this sort of house of twice the size is is worth is let us say worth two crore and the cost of entering the lottery is the cost of entering the lottery is worth one crore. So, because of this comparison it may once again you can you see that the cost of entering the lottery is actually less than the average outcome of the lottery and once again by your by the logic that is outlined above it appears that what one should do is actually enter this lottery. Now, the logic cannot be if this is your logic you cannot change it from problem to problem. However, if this is either the logic is flawed or the law or there is some way somewhere you have made a mistake in applying the logic but if you agree with this logic then the way the reason by for which you entered the lottery for the cake for the two cakes you should have also entered the lottery for the for a house of twice the size. Unfortunately, use you see most people would actually disagree that these are the same most people will not think that these this most people will say well this second the second lottery is not worth it you know there is something there is a sense in which although the above logic applies even to the second lottery there is a sense in which the second lottery seems riskier and this element of risk is what we will quantify in we will soon quantify. So, this problem the one way in which this element of risk manifest is the fact that as you scale the problem the object the outcome does not necessarily scale. So, if I replace a cake with a house if I and if I replace that 100 rupees with 1 crore the answer would not necessarily be the same. So, this is where the sort of the reasoning flaw or the reasoning inadequacy that is present in this logic and what we will see is that there is a much more a better way of looking at the problem than simply following this particular logic. This is partly from human decision making comes out also from a set of axioms and when once we go through this set of axioms it will become transparent as to why one should be adopting a certain type of model. So, with this I will pause here and I will meet you after the break and we will go over this go over a much more sophisticated way of thinking about risk.