Decision Making Under Decision Making . . . General Set - PowerPoint PPT Presentation

Need for Decision . . . From Interval to Set . . . Additivity: the Main . . . Decision Making . . . Decision Making Under Decision Making . . . General Set Uncertainty: Proof of This Result Remaining Problem Additivity Approach Main Result Proof Srialekya Edupalli and Vladik Kreinovich Home Page Title Page Department of Computer Science University of Texas at El Paso, El Paso, Texas 79968, USA, ◭◭ ◮◮ sedupalli@miners.utep.edu, vladik@utep.edu ◭ ◮ Page 1 of 22 Go Back Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 1. Need for Decision Making Under Interval Un- Additivity: the Main . . . certainty Decision Making . . . • In many practical situations, we do not know the exact Decision Making . . . consequences of different alternatives; for example: Proof of This Result Remaining Problem – we may know that investing $1000 in a project will Main Result bring us between $10 and $40 in a year, Proof – but we do not know how much exactly. Home Page • On the other hand, there are usually some alternatives Title Page with known results. ◭◭ ◮◮ • E.g., we can place this amount into a saving account ◭ ◮ at the bank. Page 2 of 22 • This will bring us exactly $20 at the end of the year. Go Back – In the first case, all we know about our gain is it is somewhere in the interval [10 , 40]. Full Screen – In the second case the gain is 20. Close Quit

Need for Decision . . . From Interval to Set . . . 2. Need for Decision Making (cont-d) Additivity: the Main . . . • Which of these two alternatives is better? Decision Making . . . Decision Making . . . • To be able to make a choice, we must be able to com- Proof of This Result pare intervals with real numbers and with intervals. Remaining Problem Main Result Proof Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 22 Go Back Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 3. From Interval to Set Uncertainty Additivity: the Main . . . • In some cases, we know that not all the values from Decision Making . . . the corresponding interval are possible. Decision Making . . . Proof of This Result • For example, we may know that we will either get $10 Remaining Problem or $40. Main Result • In this case, the set of the possible values is not the Proof whole interval [10 , 40], but the 2-point set { 10 , 40 } . Home Page • We may have more complicated situations. Title Page • For example, we may have either $10, or some value ◭◭ ◮◮ between $30 and $40. ◭ ◮ • In this case, the set of possible values is { 10 }∪ [30 , 40] . Page 4 of 22 • To make decisions in such situations, we need to com- Go Back pare sets with intervals, numbers, and other sets. Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 4. Additivity: the Main Idea Behind such Deci- Additivity: the Main . . . sion Making Decision Making . . . • Suppose that: Decision Making . . . Proof of This Result – in one situation, we have a set S 1 of possible gains Remaining Problem s 1 , and Main Result – in another independent situation, we have a set S 2 Proof of possible gains s 2 . Home Page • Then, by participating in both situation, we can gain Title Page the value s = s 1 + s 2 . ◭◭ ◮◮ • The set S of possible values of the overall gain can be ◭ ◮ obtained if we consider all possible s 1 ∈ S 1 and s 2 ∈ S 2 : Page 5 of 22 def S = S 1 + S 2 = { s 1 + s 2 : s 1 ∈ S 1 and s 2 ∈ S 2 } . Go Back • It is reasonable to assign, to each set S , the price u ( S ) Full Screen we can pay to participate in this situation. Close Quit

Need for Decision . . . From Interval to Set . . . 5. Additivity (cont-d) Additivity: the Main . . . • If the sets S 1 , S 2 have the same price ( u ( S 1 ) = u ( S 2 )), Decision Making . . . we say that these two sets are equivalent : S 1 ≡ S 2 . Decision Making . . . Proof of This Result • The price to participate in both events should be equal Remaining Problem to the sum of the prices: u ( S 1 + S 2 ) = u ( S 1 ) + u ( S 2 ) . Main Result • This property is known as additivity . Proof • Let S be a class of sets which is closed under addition. Home Page • An equivalence relation ≡ is called additive if: Title Page if S 1 + S 2 = S ′ 1 + S 2 then S 1 ≡ S ′ 1 . ◭◭ ◮◮ ◭ ◮ def • For every additive function u , the relation S 1 ≡ S 2 = ( u ( S 1 ) = u ( S 2 )) is additive. Page 6 of 22 • Indeed, if S 1 + S 2 = S ′ Go Back 1 + S 2 , then, due to additivity, u ( S 1 ) + u ( S 2 ) = u ( S ′ 1 ) + u ( S 2 ) . Full Screen • Thus, u ( S ′ 1 ) = u ( S 1 ) and S ′ 1 ≡ S 1 . Close Quit

Need for Decision . . . From Interval to Set . . . 6. Decision Making Under Interval Uncertainty: Additivity: the Main . . . What Is Known Decision Making . . . • In case the set of possible gains is an interval [ a, a ], no Decision Making . . . matter what happens, we will get ≥ a and ≤ a . Proof of This Result Remaining Problem • Thus, the price of this interval cannot be lower than a Main Result and cannot be higher than a . Proof • We say that a real-valued function u defined on the set Home Page of all intervals is consistent if for each interval, we have Title Page a ≤ u ([ a, a ]) ≤ a. ◭◭ ◮◮ • Every consistent additive function u on the set of all ◭ ◮ intervals has the form Page 7 of 22 u ([ a, a ]) = α · u + (1 − α ) · u, for some α ∈ [0 , 1] . Go Back Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 7. Hurwicz Criterion Additivity: the Main . . . • This formula was first proposed by the future Nobel Decision Making . . . prize winner Leo Hurwicz. Decision Making . . . Proof of This Result • It is known as Hurwicz optimism-pessimism criterion . Remaining Problem • Optimism corresponds to α = 1, when a decision maker Main Result values the interval as much as its largest value. Proof Home Page • So, in effect, he/she considers only the best value from this interval to be possible. Title Page • Pessimism corresponds to α = 0, when a decision maker ◭◭ ◮◮ values the interval as much as its smallest value. ◭ ◮ Page 8 of 22 Go Back Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 8. Decision Making Under Set Uncertainty: What Additivity: the Main . . . Is Known Decision Making . . . • It is known how to make a decision when S is bounded Decision Making . . . and closed (i.e., contains all its limit points). Proof of This Result Remaining Problem • For every additive equivalence relation on the class of Main Result all bounded closed sets, S ≡ [inf( S ) , sup( S )] . Proof • Thus, the utility of each set S is equal to the utility of Home Page the corresponding interval [inf( S ) , sup( S )] . Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 22 Go Back Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 9. Proof of This Result Additivity: the Main . . . • Every bounded closed sets contains its limit points; in Decision Making . . . particular, it contains the points inf( S ) and sup( S ). Decision Making . . . Proof of This Result • Thus, { inf( S ) , sup( S ) } ⊆ S ⊆ [inf( S ) , sup( S )] . Remaining Problem • So, by a clear set-inclusion monotonicity of set addi- Main Result tion, we conclude that Proof Home Page { inf( S ) , sup( S ) } +[inf( S ) , sup( S )] ⊆ S +[inf( S ) , sup( S )] ⊆ Title Page [inf( S ) , sup( S )] + [inf( S ) , sup( S )] . ◭◭ ◮◮ • However, one can easily check that ◭ ◮ { inf( S ) , sup( S ) } + [inf( S ) , sup( S )] = Page 10 of 22 [inf( S ) , sup( S )]+[inf( S ) , sup( S )] = [2 inf( S ) , 2 sup( S )] . Go Back Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 10. Proof of This Result (cont-d) Additivity: the Main . . . • Thus, the intermediate set S + [inf( S ) , sup( S )] should Decision Making . . . be equal to the same interval: Decision Making . . . Proof of This Result S +[inf( S ) , sup( S )] = [inf( S ) , sup( S )]+[inf( S ) , sup( S )] = Remaining Problem [2 inf( S ) , 2 sup( S )] . Main Result Proof • Since the equivalence relation is assumed to be addi- Home Page tive, we conclude that S ≡ [inf( S ) , sup( S )] . Title Page • The proposition is proven. ◭◭ ◮◮ ◭ ◮ Page 11 of 22 Go Back Full Screen Close Quit

Need for Decision . . . From Interval to Set . . . 11. Remaining Problem Additivity: the Main . . . • Boundedness is reasonable: in all real-life situations, Decision Making . . . we have lower and upper bounds on possible gains: Decision Making . . . Proof of This Result – in usual investments, we do not expect to gain mil- Remaining Problem lions, and Main Result – we do not exact to lose millions – since usually, we Proof just do not have these millions to lose. Home Page • However, the requirement that the set be closed may Title Page be too restrictive; for example: ◭◭ ◮◮ – we may know that the gain will be between 0 and ◭ ◮ $100, Page 12 of 22 – but we are sure that the gain cannot be zero and cannot be exactly $100. Go Back Full Screen Close Quit