Lecture 19

[Music] welcome back my dear friends a very good morning good afternoon and good evening to all of you at whichever time you are hearing these lectures and whichever part of the world you are and this course title is multicriteria decision making under NP Muk series which is total of 60 lectures and spread over 12 weeks and each week we have five lectures and each lecture being for half an hour and my good name is raghunandan s Gupta from the IM department at I kpur in India so this is the 19th lecture which me means we are in the almost end of the fourth week and the broader set of topics which we have been continuing for quite long time is basically about the mcdm definitions concepts of utility Theory uh just to recap because we going to end this broader set of topics but just I'll just mention the important points the concept of utility Theory how you find out this expected value and there were many examples you are discussed then we considered different concepts of the properties four properties non ciation concept of risk concept of absolute rision relative ravion then we considered the four different types of utility functions with their properties related to A and R separately for all these four four were quadratic the exponential power and logarithmic then we consider an important concept of certainty equivalent and how certainty equivalent can be used then we went into the ideas of trying to find utilize certainty equivalent to map how the util function of a person looks and then we further on went on to geometric main methods and based on that how you can rank decisions then we found out the equivalence between normality uh of the returns and their concept of quadratic uty function then we left utility went into Safety First principle and consider the three principles of Safety First principle one is minimizing the probability of RL RP being less than RL RL is the bound it can be replaced by RF also and then the second was maximizing of RL maximizing of RP bar so in this lecture we will consider stochastic dominance and the concept concept of hyperbolic absolute dision property so hopefully I can go into hyperbolic absolute dision property or else it will be continued in the 20th lecture so I'll come to the examples of stochastic dominance first order second order within few minutes but first the concept maybe it's mathematical but lay let us lay stress on that consider two random variables X1 and X2 it can a and b also and with the respective probability density function and but we for AR examples we consider very simply the ideas of probability Mass function the ideas of of trying to convey using examples would be from probability Mass function but it can be replicated for a continuous distribution also with the respective PDF and the distribution function distribution function is basically the cumulative distribution functions they are given for the first random variable which is X1 I'll mark with the color blue so X1 random variable PDF small ff1 X1 and CDF of capital F1 X1 for the second random variable which I'll mark in in see for example uh green let me use the dark green which is X2 for the second random variable the PDF is small F2 X2 and the CDF is capital F2 X2 then by definition we will consider the following properties the zeroth order dominance of X1 over X2 will be applicable if and only if X1 is less than equal to X2 for all realization of the random variables and X1 will be strictly less than X2 for at least one realization so if I am trying to plot I'll come to the examples as I M for the for the first order and second order very soon with concrete examples so if I have the real line and if I find out the values of X1 I'll denote by Blue so these are the D variables as you pick up they are on the real line and I'm choosing the corresponding values of x two which are in green so if there are more strictly greater some cases they are equal to so that means both the values should coincide which I'm marking here so in that case you have the zeroth order dominance of X1 over X2 X1 is the blue one remember so let me denote X1 and X2 was the green one sorry the color didn't change this is X2 now as I said we'll come to the first order and second order dominance sastic dominance using two random variables X1 and X2 and consider for the example now the first order dominance is true if and only if of X1 / X2 if the probability of X1 being greater than y y is some value is less than equal to probability of X2 being greater than y for all y's on the real line as that there exist at least one y such that rather than less than equal to the less than in the strict sense prevails now what I want you to understand is this if I consider the probability is less than in the sense let me write this so if probability of X1 - Y is less than equal to probability of X2 is greater than y this is greater than so they are greater means onto the right hand side if I'm looking at the real line from my side so random variable will increases onto the right it decrease onto the left so probability being greater than x Y which means that 1 minus that would be onto the left hand side so I basically put 1 minus first if I when I put the minus sign this inequality would change so the inequality for a minus sign so minus a minus it becomes greater than equal to then I add one plus onto the left hand side A plus one onto the right hand side also so based on that when I find out it will be the probability here now it's a greater than sign so it means the probability of X1 being less than y not greater than because I've put the inequality and put added a one and first I change the sign would be greater than probability of X2 being less than equal to Y which means if I draw this diagram uh let me make a blank slide and then do it accordingly so this part I'll again do it in the blank side I'll discard it and WR so it's easier for us to understand by the way the blank slide which I'm going to create would not be there in the slides which are uploaded they would be there in the video so you can understand because the lectures slides are written systematically any extra portion I'm just writing the blank slide why I'm creating because the space is limited as you remember in the earlier class which I did so here I will write the probability I'll again write the small state of definitions and derivations so it's easy so probability of X1 so this was the case for the first order squ so I make it minus s diverses minus sign I add one greater than sign the remains so 1 minus basically now becomes probability what does this mean the last statement which I'm putting a red arrow so if I consider the distribution consider the real line there are two distributions I denote one by green and which is say for example X1 and the another one as blue which is the y1 which is X2 and this is the mean value this is the mean value so what I want to ensure is there is some value of y which I'll denote by Red and consider I'm taking y as as see for example fixed it can change is the overall probability of X1 being less than y onto the left side and which would always be more than the probability of X2 being less than y the probability of sense it you are trying to cover more the second order dominance would mean when I find so this first order of dominance basically does with the distribution function the second order dominance means for X1 being greater than X2 if only if if the integration of the whole area of the of the cumulative functions of X2 is greater than y minus probability of X1 is greater than y is always positive for all Y and this greater than equal to Z is strictly greater than Z for some y that second order dominance would hold true so at least one in both the first cases should be greater and the second shade should be greater not greater than equal to in both the cases would give you the actual concept of first order second order correspondingly now few conditions I will state without proof the first order dominance stochastic dominance of X1 / X2 is a sufficient condition for the second order of X1 or X2 so obviously if I have the first order second order third order so on and so forth this concept of sufficiency conditions for first to second second to Third and higher orders can be made accordingly the second point is X2 is a mean preserving spread of X1 then if that is true then X1 will be second order dominant ated over X2 and what we mean by mean preserving spread is basically a change from one probability distribution to the other consider from A to B where B is formed by spreading out because the probability total sum is one so what we do is that by spreading out B by forming out one or more portions of A's probability density function says that leaving the mean value unchange we just spread it out more so to make things easier let us consider two different examples diagramatically I'm not going to use any equations the red one which you see is the uniform continuous case distribution between Min -2 and +2 and obviously if you know the mean value is given by a + b B by 2 and if I consider these values for the red the expected value zero as shown here by the black vertical line where I'll put a red tick mark so this is for the red one which is X1 distributed uniform - 2 2 + 2 now consider the distribution X2 with the same expected value which is also the vertical black line and I'll use the dark blue to denote X2 and it is spread between minus 3 and + 3 so X2 is distributed uniform - 3 to + 3 again the average value would be zero so zero is fixed mean value is fixed for B is much more spread out if you can see spread out more while the first one so this horizontal lines are zero so actually the distribution is this so this would be the concept of means preserving spreads and if X2 is a mean preserving spread of X1 as in the diagram where you have the uniform distribution of - 3 to + 3 with respect to that X1 which is the first distribution then X1 will dominate x2 in the second order sense let us consider a second diagrammatic representation the first one is basically in the blue color is for standard normal deviate X1 which is distributed if I use the blue BL color X1 is distributed normal with 01 and this is the distribution which you see here s and X2 if I consider such that both the mean values of X1 and X2 are same because they are given as zero so X2 is normally distributed with 0 2 and the distribution is I'll just mark it once this is the distribution so if X2 is the mean preserving of X1 then X1 strictly in the second order sense would dominate X2 now the question is that um just for your information you saw the examples where the mean values were zero so is it necessary for the mean values to be zero no I could have basically taken any other distribution uh for about the case of discrete and the normal case with different means so if I consider this example maybe I could have considered X1 as normal see for example three V standard division of two or variance of 2 squ and X2 can be the second distribution normal distribution ated with three also but the spread is more consider it is now four standard division or 16 is the variance so this can be also done similarly for the first example I need not choose the mean value as zero it could have been say for example X1 uniform between see for example - 10 to -5 so obviously the average value would be what - 10 - 5 by 2 is = to - 7.5 similarly I could have X2 done accordingly which which is uniform and in that case I want to ensure minus 7.5 is the value so I could have had say for example minus 12 and uh the values would be see for example if it is -2 and - 3 so the value is -12 - 3 by 2 is basically -5 by 2 is equal to - 7.5 but if in the second case which is the Violet one the spread is much more in the rate case it is much more concentrated so the values would give you the idea of mean preserving spread such that X1 will dominate x2 in the case where the example was the discrete distribution and for the case when the distribution is considered as normal I just thought these two examples would be good one to illustrate now till now I have said mentioned about stochastic dominance but I've have not discussed the examples so these are the examples one by one for the stochastic dominance for the first order and the second order so let us consider the example as such so there are states of nature which we denote by 1 2 3 the outcomes what is the probabilities and there are two different decisions decisions I'll denote by A and B now the amounts of the outcome for the states I have taken arbitary values they can be changed and you can formulate different problems to illustrate that other idea is that question would be that states of nature is only one to three can there be more yes they can be more they can be less also they can be two outcomes or they can be four five six outcomes also and decisions I have taken as a they can be more more than two also like a b c d accordingly but I've made it simple the amounts are given for state of natur as 10 30 70 and as per the decision a and decision B the corresponding probabilities for decision a for the states of nature of 1 2 3 are I'll mark with a different color see for example red the probabilities are 4.60 and you see the probability sums up to one obviously it has to be decision a is out of the state of nature only there cannot be anything outside similarly if I consider decision B I'll use a green color the corresponding probabilities are 3.3.4 again the sum is 1 now I need to find out the expected value first I'm just giving so the expected value for decision a would be what 10 into point4 + 30 into. 6 + 17 to 0 because the last probability is z the value comes out to be using the same coloring scheme comes out to be 22 and when I use the expected value formula for decision B the corresponding values are which I sum them up is 10 into. 3 + 30 into 3 + 77 into point4 and the value comes out to be 40 now let us draw the step functions of the decisions accordingly along with the expected value for decision a and decision B corresponding to the states of nature 1 2 3 separately and then I merge them I've used different colors so that you can understand them now consider fub1 is for X1 which is blue in color and FS2 which is X2 is red in color so if I consider the corresponding probabilities so this consider only on the blue line so if I consider the Blue Line the jumps are accordingly jumps are how if I consider the decision a it is4 first and then the jump happens 6 so the cumulative values are 4 and 1 where is point4 is happening point4 is happening as the state of nature amount is one n amount is 10 state of nature is one and the second gy is happening which becomes the cumulative distribution function becomes one is happening at when the state of nature is two where the amount is 30 so here it is 1030 4.6 let us see for the blue one it is 0 till 10 and there is a jump so the jump happens of4 if you remember so the jump happens here I should use so this is the I should use the red color um black one to denote so this is the first jump of point 4 then it is constant constant then when the value 30 occurs it jumps by6 and it becomes one so the blue one is basically for X1 which is decision a now consider let us consider decision B what are the jumps jumps are points 3.3.4 at 10370 so at 10 it becomes 3 which is the red one continues till 30 again jumps 3 becomes 6 continues still 70 and then jumps 4 so 3.3.4 now the interesting part is two things number one we see the expected value of X2 which is 40 which we calculated and the expected value of X1 decision a is 22 but interestingly throughout the distribution X1 which is the blue one always dominates and how I'm saying it dominates because wherever you see in the graph x the blue line for X1 is always above the X1 X2 so which you see here I'll just Market this portion obviously it is more again this portion is more so there's no portion of X1 which is blue one is below the red one it can be equal in many of the cases we can formulate the problem but it would never be fall below the X1 if X1 strictly dominates or dominates x2 in the first order sense and obviously if I'm going to consider in the strict sense not in the greater than equal to sign in the strict sense for X1 dominating X2 I'll draw the graph I didn't draw the theoretical one because for the Practical one it will be easy for you to understand if I draw the graph consider the aises are AIS is given where the axis are along the X you can I'm plotting the amounts as usual or the decisions based on which I'm trying to find and if I plot X1 like this because if you remember the probability is always one it cannot exceed and X2 as the blue one ultimately the probability will be one in this case the red one which will will denote by X1 with respect to the blue one which will denote by X2 X1 strictly dominates X2 throughout because there's no portion it is equal equal to sign if you remember would be for the case when there is not strictly dominating so with this I will end this 19th lecture and continue discussing about the concept of stochastic dominance further on thank you very much and have a nice [Music] [Music] [Music] day