Decision Making . . . Decision Under . . . Question How to Estimate Hurwicz Our Result Value of the Union of Two Our Result (cont-d) References Intervals If We Know the Home Page Hurwicz Values for Both Title Page Intervals? ◭◭ ◮◮ ◭ ◮ Mahdokht Afravi, Gerardo Cervantes, Xavier Martinez, Matthew Melvin, Victor Vargas, Ana Zepeda, Page 1 of 7 and Vladik Kreinovich Go Back Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA, mafravi@miners.utep.edu, Full Screen gcervantes8@miners.utep.edu, xamartinez2@miners.utep.edu, mjmelvin@miners.utep.edu, vevargascor@miners.utep.edu, Close alzepeda@miners.utep.edu, vladik@utep.edu Quit
1. Decision Making Under Interval Uncertainty Decision Making . . . Decision Under . . . • The decision maker’s preferences can be described by Question the corresponding utility function : Our Result – the larger its value, Our Result (cont-d) – the better the alternative. References • In practice, we often do not know the exact values of Home Page the utility corresponding to each alternative. Title Page • Instead, we only know bounds u and u the utility u . ◭◭ ◮◮ • In other words, we know that u is in the interval ◭ ◮ u = [ u, u ] . Page 2 of 7 • In such situations, decision theory recommends select- Go Back ing the alternative with the largest value of Full Screen H ( u ) = α · u + (1 − α ) · u. Close • This expression is known as Hurwicz value . Quit
Decision Making . . . 2. Decision Under Interval Uncertainty (cont-d) Decision Under . . . • Reminder: H ( u ) = α · u + (1 − α ) · u. Question Our Result • The parameter α describes the optimism level of the Our Result (cont-d) decision maker; e.g.: References – the value α = 1 corresponds to perfect optimism, while Home Page – the value α = 0 corresponds to perfect pessimism. Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 7 Go Back Full Screen Close Quit
3. Question Decision Making . . . Decision Under . . . • In some cases, an action can result in two possible out- Question comes. Our Result • For each outcome, we know the utility interval u 1 and u 2 . Our Result (cont-d) References • We do not know which outcome will happen. Home Page • So, overall, the set of possible utility values is the union Title Page u 1 ∪ u 2 of these intervals. ◭◭ ◮◮ • To be precise, in general, it is the interval hull [min( u 1 , u 2 ) , max( u 1 , u 2 )] of this union. ◭ ◮ • If we know Hurwicz values of the two intervals, what Page 4 of 7 can we say about the Hurwicz value of their union? Go Back Full Screen Close Quit
4. Our Result Decision Making . . . Decision Under . . . • Our result is that Question Our Result min( H ( u 1 ) , H ( u 2 )) ≤ H ( u 1 ∪ u 2 ) ≤ max( H ( u 1 ) , H ( u 2 )) . Our Result (cont-d) • Without losing generality, we will prove only the right References inequality. Home Page • We consider two possible cases: Title Page – the case when the order between the bounds in the ◭◭ ◮◮ same for both bounds, i.e., ◭ ◮ ∗ either u 1 ≤ u 2 and u 1 ≤ u 2 , Page 5 of 7 ∗ or u 2 ≤ u 1 and u 2 ≤ u 1 ; Go Back – and the case when the order between lower bounds is different from the order between upper bounds. Full Screen Close Quit
Decision Making . . . 5. Our Result (cont-d) Decision Under . . . • In the first case, without losing generality, we can as- Question sume that u 1 ≤ u 2 and u 1 ≤ u 2 . Our Result • Then, the union has the form [ u 1 , u 2 ]. Our Result (cont-d) References • Here, H ( u 2 ) = α · u 2 + (1 − α ) · u 2 and H ( u 1 ∪ u 2 ) = α · u 2 + (1 − α ) · u 1 . Home Page • Since u 1 ≤ u 2 , we conclude that H ( u 1 ∪ u 2 ) ≤ H ( u 2 ) Title Page and thus, ◭◭ ◮◮ H ( u 1 ∪ u 2 ) ≤ max( H ( u 1 ) , H ( u 2 )) . ◭ ◮ Page 6 of 7 • In the second case, without losing generality, we can assume that u 1 ≤ u 2 and u 2 ≤ u 1 . Go Back • In this case, min( u 1 , u 2 ) = u 1 and max( u 1 , u 2 ) = u 1 . Full Screen Close • Thus u 1 ∪ u 2 = u 1 hence H ( u 1 ∪ u 2 ) = H ( u 1 ) and thus too, H ( u 1 ∪ u 2 ) ≤ max( H ( u 1 ) , H ( u 2 )) . Quit
6. References Decision Making . . . Decision Under . . . • L. Hurwicz, Optimality Criteria for Decision Making Question Under Ignorance , Cowles Commission Discussion Pa- Our Result Our Result (cont-d) per, Statistics, No. 370, 1951. References • V. Kreinovich, “Decision making under interval un- certainty (and beyond)”, In: P. Guo and W. Pedrycz Home Page (eds.), Human-Centric Decision-Making Models for So- Title Page cial Sciences , Springer Verlag, 2014, pp. 163–193. ◭◭ ◮◮ • D. Luce and R. Raiffa, Games and Decisions: Intro- ◭ ◮ duction and Critical Survey , Dover, New York, 1989. Page 7 of 7 Go Back Full Screen Close Quit
Recommend
More recommend
