Decision Making . . . Hurwicz’s Idea Scale Invariance Additivity Decision Making Under Need to Go Beyond . . . Interval Uncertainty: Analysis of the Problem Scale Invariance and . . . Beyond Hurwicz Main Result Relation to Non- . . . Pessimism-Optimism Home Page Criterion Title Page ◭◭ ◮◮ Tran Anh Tuan 1 , Vladik Kreinovich 2 , and Thach Ngoc Nguyen 3 ◭ ◮ Page 1 of 25 1 Ho Chi Minh City Institute of Development Studies, Vietnam, at7tran@gmail.com Go Back 2 University of Texas at El Paso, El Paso, Texas 79968, USA vladik@utep.edu Full Screen 3 Banking University of Ho Chi Minh City, Vietnam, Close Thachnn@buh.edu.vn Quit
Decision Making . . . Hurwicz’s Idea 1. Decision Making under Interval Uncertainty Scale Invariance • In the ideal case, we know the exact consequence of Additivity each action. Need to Go Beyond . . . Analysis of the Problem • In this case, a natural idea is to select an action that Scale Invariance and . . . will lead to the largest profit. Main Result • In real life, we rarely know the exact consequence of Relation to Non- . . . each action. Home Page • In many cases, all we know are the lower and upper Title Page bound on the quantities describing such consequences. ◭◭ ◮◮ • So, all we know is an interval [ u, u ] that contains the ◭ ◮ actual (unknown) value u . Page 2 of 25 • So, we have several alternatives a for each of which we Go Back only have an interval estimate [ u ( a ) , u ( a )]. Full Screen • Which alternative should we select? Close Quit
Decision Making . . . Hurwicz’s Idea 2. Hurwicz’s Idea Scale Invariance • The problem of decision making under interval uncer- Additivity tainty was first handled by a Nobelist Leo Hurwicz. Need to Go Beyond . . . Analysis of the Problem • Hurwicz’s main idea was as follows. Scale Invariance and . . . • We know how to make decisions when for each alter- Main Result native, we know the exact value of the resulting profit. Relation to Non- . . . Home Page • So, to help decision makers make decisions under in- terval uncertainty, Hurwicz proposed: Title Page – to assign, to each interval a = [ a, a ], an equivalent ◭◭ ◮◮ value u H ( a ), and ◭ ◮ – then to select an alternative with the largest equiv- Page 3 of 25 alent value. Go Back Full Screen Close Quit
Decision Making . . . Hurwicz’s Idea 3. Natural Requirements on u H ( a ) Scale Invariance • Of course, when we know the exact consequence a , we Additivity should have u H ([ a, a ]) = a . Need to Go Beyond . . . Analysis of the Problem • All the values a from the interval [ a, a ] are larger than Scale Invariance and . . . (thus better than) or equal to the lower endpoint a . Main Result • So, the equivalent value must also be larger than or Relation to Non- . . . equal to a . Home Page • Similarly, all the values a from the interval [ a, a ] are Title Page worse than or equal to the upper endpoint a . ◭◭ ◮◮ • So, the equivalent value must also be smaller than or ◭ ◮ equal to a : a ≤ u H ([ a, a ]) ≤ a. Page 4 of 25 Go Back Full Screen Close Quit
Decision Making . . . Hurwicz’s Idea 4. Scale Invariance Scale Invariance • The equivalent value should not change if we change a Additivity monetary unit. Need to Go Beyond . . . Analysis of the Problem • What was better when we count in dollars should also Scale Invariance and . . . be better when we use Vietnamese Dongs instead. Main Result • We can change from the original monetary unit to a Relation to Non- . . . new unit which is k times smaller. Home Page • Then, all the numerical values are multiplied by k . Title Page • Thus, if we have u H ( a, a ) = a 0 , then, for all k > 0, we ◭◭ ◮◮ should have u H ([ k · a, k · a ]) = k · a 0 . ◭ ◮ Page 5 of 25 Go Back Full Screen Close Quit
Decision Making . . . Hurwicz’s Idea 5. Additivity Scale Invariance • Suppose that we have two separate independent situa- Additivity tions, with possible profits [ a, a ] and [ b, b ]s. Need to Go Beyond . . . Analysis of the Problem • Then, the overall profit of these two situations can take Scale Invariance and . . . any value from a + b to a + b . Main Result • So, we can compute the equivalent value of the corre- Relation to Non- . . . sponding interval Home Page def a + b = [ a + b, a + b ] . Title Page • Second, we can first find equivalent values of each of ◭◭ ◮◮ the intervals and then add them up. ◭ ◮ • It is reasonable to require that the resulting value should Page 6 of 25 be the same in both cases, i.e., that we should have Go Back u H ([ a + b, a + b ]) = u H ([ a, a ]) + h H ([ b, b ]) . Full Screen • This property is known as additivity . Close Quit
Decision Making . . . Hurwicz’s Idea 6. Derivation of Hurwicz Formula Scale Invariance def • Let us denote α H = u H ([0 , 1]); due to the first natural Additivity requirement, 0 ≤ α H ≤ 1. Need to Go Beyond . . . Analysis of the Problem • Now, due to scale-invariance, for every value a > 0, we Scale Invariance and . . . have u H ([0 , a ]) = α H · a . Main Result • For a = 0, this is also true, since in this case, we have Relation to Non- . . . u H ([0 , 0]) = 0 . Home Page Title Page • In particular, for every two values a ≤ a , we have ◭◭ ◮◮ u H ([0 , a − a ]) = α H · ( a − a ) . ◭ ◮ • Now, we also have u H ([ a, a ]) = a . Page 7 of 25 • Thus, by additivity, we get u H ([ a, a ]) = ( a − a ) · α H + a, Go Back i.e., u H ([ a, a ]) = α H · a + (1 − α H ) · a. Full Screen • This is the formula for which Leo Hurwicz got his Nobel prize. Close Quit
Decision Making . . . Hurwicz’s Idea 7. Meaning of Hurwicz Formula Scale Invariance • When α H = 1, this means that the equivalent value is Additivity equal to the largest possible value a . Need to Go Beyond . . . Analysis of the Problem • So, when making a decision, the person only takes into Scale Invariance and . . . account the best possible scenario. Main Result • In real life, such a person is known as an optimist . Relation to Non- . . . • When α H = 0, this means that the equivalent value is Home Page equal to the smallest possible value a . Title Page • So, when making a decision, the person only takes into ◭◭ ◮◮ account the worst possible scenario. ◭ ◮ • In real life, such a person is known as an pessimist . Page 8 of 25 • When 0 < α H < 1, this means that a person takes into Go Back account both good and bad possibilities. Full Screen • So, α H is called optimism-pessimism coefficient , and the procedure optimism-pessimism criterion. Close Quit
Decision Making . . . Hurwicz’s Idea 8. Need to Go Beyond Hurwicz Criterion Scale Invariance • While Hurwicz criterion is reasonable, it leaves several Additivity options equivalent which should not be equivalent. Need to Go Beyond . . . Analysis of the Problem • For example, if α H = 0 . 5, then, according to Hurwicz Scale Invariance and . . . criterion, the interval [ − 1 , 1] should be equivalent to 0. Main Result • However, a risk-averse person will prefer status quo (0) Relation to Non- . . . to a situation [ − 1 , 1] in which s/he can lose. Home Page • A risk-prone person would prefer [ − 1 , 1] in which he/she Title Page can gain. ◭◭ ◮◮ • To take this into account, we need to go beyond as- ◭ ◮ signing a numerical value to each interval. Page 9 of 25 • We need, instead, to describe possible orders on the Go Back class of all intervals. Full Screen • This is what we do in this talk. Close Quit
Decision Making . . . Hurwicz’s Idea 9. Analysis of the Problem Scale Invariance • For every two alternatives a and b , we want to provide Additivity one of the following three recommendations: Need to Go Beyond . . . Analysis of the Problem – select the first alternative; we will denote this rec- Scale Invariance and . . . ommendation by b < a ; Main Result – select the second alternative; we will denote this Relation to Non- . . . recommendation by a < b ; or Home Page – treat these two alternatives as equivalent ones; we will denote this recommendation by a ∼ b . Title Page ◭◭ ◮◮ • Our recommendations should be consistent: e.g., ◭ ◮ – if we recommend that b is preferable to a and that c is preferable to b , Page 10 of 25 – then we should also recommend that c is preferable Go Back to a . Full Screen • Such consistency can be described by the following def- Close inition. Quit
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