Decision Making Case of Interval . . . under Interval Monetary - PowerPoint PPT Presentation

Need for Decision . . . When Monetary . . . Hurwicz Optimism- . . . Fair Price Approach: . . . Decision Making Case of Interval . . . under Interval Monetary Approach Is . . . The Notion of Utility (and More General) Group Decision . . . We Must Take . . . Uncertainty: Home Page Monetary vs. Utility Title Page Approaches ◭◭ ◮◮ ◭ ◮ Vladik Kreinovich Page 1 of 72 University of Texas at El Paso El Paso, TX 79968, USA Go Back vladik@utep.edu Full Screen Close Quit

Need for Decision . . . When Monetary . . . 1. Need for Decision Making Hurwicz Optimism- . . . • In many practical situations: Fair Price Approach: . . . Case of Interval . . . – we have several alternatives, and Monetary Approach Is . . . – we need to select one of these alternatives. The Notion of Utility • Examples: Group Decision . . . We Must Take . . . – a person saving for retirement needs to find the best Home Page way to invest money; Title Page – a company needs to select a location for its new plant; ◭◭ ◮◮ – a designer must select one of several possible de- ◭ ◮ signs for a new airplane; Page 2 of 72 – a medical doctor needs to select a treatment for a Go Back patient. Full Screen Close Quit

Need for Decision . . . When Monetary . . . 2. Need for Decision Making Under Uncertainty Hurwicz Optimism- . . . • Decision making is easier if we know the exact conse- Fair Price Approach: . . . quences of each alternative selection. Case of Interval . . . Monetary Approach Is . . . • Often, however: The Notion of Utility – we only have an incomplete information about con- Group Decision . . . sequences of different alternative, and We Must Take . . . – we need to select an alternative under this uncer- Home Page tainty. Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 72 Go Back Full Screen Close Quit

Need for Decision . . . When Monetary . . . 3. When Monetary Approach Is Appropriate Hurwicz Optimism- . . . • In many situations, e.g., in financial and economic de- Fair Price Approach: . . . cision making, the decision results: Case of Interval . . . Monetary Approach Is . . . – either in a money gain (or loss) and/or The Notion of Utility – in the gain of goods that can be exchanged for Group Decision . . . money or for other goods. We Must Take . . . • In this case, we select an alternative which the highest Home Page exchange value, i.e., the highest price u . Title Page • Uncertainty means that we do not know the exact ◭◭ ◮◮ prices. ◭ ◮ • The simplest case is when we only know lower and Page 4 of 72 upper bounds on the price: u ∈ [ u, u ]. Go Back Full Screen Close Quit

Need for Decision . . . When Monetary . . . 4. Hurwicz Optimism-Pessimism Approach to De- Hurwicz Optimism- . . . cision Making under Interval Uncertainty Fair Price Approach: . . . • L. Hurwicz’s idea is to select an alternative s.t. Case of Interval . . . Monetary Approach Is . . . α H · u + (1 − α H ) · u → max . The Notion of Utility • Here, α H ∈ [0 , 1] described the optimism level of a Group Decision . . . decision maker: We Must Take . . . Home Page • α H = 1 means optimism; Title Page • α H = 0 means pessimism; ◭◭ ◮◮ • 0 < α H < 1 combines optimism and pessimism. ◭ ◮ + This approach works well in practice. Page 5 of 72 − However, this is a semi-heuristic idea. Go Back ? It is desirable to come up with an approach which can Full Screen be uniquely determined based first principles. Close Quit

Need for Decision . . . When Monetary . . . 5. Numerical Example Hurwicz Optimism- . . . • Suppose that we have two alternatives: Fair Price Approach: . . . Case of Interval . . . – one in which we gain $1,000 for sure, and Monetary Approach Is . . . – one in which we may gain $2,500, but may gain The Notion of Utility nothing, and Group Decision . . . – we have no information about the probabilities of We Must Take . . . different gains. Home Page • Which option should we choose? Title Page • An optimist chooses the second alternative. ◭◭ ◮◮ • A pessimist chooses the first alternative. ◭ ◮ • For α = 0 . 5, the second alternative is better: Page 6 of 72 α · u + (1 − α ) · u = 0 . 5 · 2500 + 0 . 5 · 0 = 1250 > 1000 . Go Back Full Screen • In general, for α > 0 . 4, the second alternative is better, otherwise the first one. Close Quit

Need for Decision . . . When Monetary . . . 6. Fair Price Approach: An Idea Hurwicz Optimism- . . . • When we have a full information about an object, then: Fair Price Approach: . . . Case of Interval . . . – we can express our desirability of each possible sit- Monetary Approach Is . . . uation The Notion of Utility – by declaring a price that we are willing to pay to Group Decision . . . get involved in this situation. We Must Take . . . • Once these prices are set, we simply select the alterna- Home Page tive for which the participation price is the highest. Title Page • In decision making under uncertainty, it is not easy to ◭◭ ◮◮ come up with a fair price. ◭ ◮ • A natural idea is to develop techniques for producing Page 7 of 72 such fair prices. Go Back • These prices can then be used in decision making, to Full Screen select an appropriate alternative. Close Quit

Need for Decision . . . When Monetary . . . 7. Case of Interval Uncertainty Hurwicz Optimism- . . . • Ideal case: we know the exact gain u of selecting an Fair Price Approach: . . . alternative. Case of Interval . . . Monetary Approach Is . . . • A more realistic case: we only know the lower bound The Notion of Utility u and the upper bound u on this gain. Group Decision . . . • Comment: we do not know which values u ∈ [ u, u ] are We Must Take . . . more probable or less probable. Home Page • This situation is known as interval uncertainty . Title Page • We want to assign, to each interval [ u, u ], a number ◭◭ ◮◮ P ([ u, u ]) describing the fair price of this interval. ◭ ◮ • Since we know that u ≤ u , we have P ([ u, u ]) ≤ u . Page 8 of 72 • Since we know that u ≤ u , we have u ≤ P ([ u, u ]). Go Back Full Screen Close Quit

Need for Decision . . . When Monetary . . . 8. Case of Interval Uncertainty: Monotonicity Hurwicz Optimism- . . . • Case 1: we keep the lower endpoint u intact but in- Fair Price Approach: . . . crease the upper bound. Case of Interval . . . Monetary Approach Is . . . • This means that we: The Notion of Utility – keeping all the previous possibilities, but Group Decision . . . – we allow new possibilities, with a higher gain. We Must Take . . . Home Page • In this case, it is reasonable to require that the corre- sponding price not decrease: Title Page ◭◭ ◮◮ if u = v and u < v then P ([ u, u ]) ≤ P ([ v, v ]) . ◭ ◮ • Case 2: we dismiss some low-gain alternatives. Page 9 of 72 • This should increase (or at least not decrease) the fair Go Back price: Full Screen if u < v and u = v then P ([ u, u ]) ≤ P ([ v, v ]) . Close Quit

Need for Decision . . . When Monetary . . . 9. Additivity: Idea Hurwicz Optimism- . . . • Let us consider the situation when we have two conse- Fair Price Approach: . . . quent independent decisions. Case of Interval . . . Monetary Approach Is . . . • We can consider two decision processes separately. The Notion of Utility • We can also consider a single decision process in which Group Decision . . . we select a pair of alternatives: We Must Take . . . Home Page – the 1st alternative corr. to the 1st decision, and – the 2nd alternative corr. to the 2nd decision. Title Page ◭◭ ◮◮ • If we are willing to pay: ◭ ◮ – the amount u to participate in the first process, and Page 10 of 72 – the amount v to participate in the second decision process, Go Back • then we should be willing to pay u + v to participate Full Screen in both decision processes. Close Quit

Need for Decision . . . When Monetary . . . 10. Additivity: Case of Interval Uncertainty Hurwicz Optimism- . . . • About the gain u from the first alternative, we only Fair Price Approach: . . . know that this (unknown) gain is in [ u, u ]. Case of Interval . . . Monetary Approach Is . . . • About the gain v from the second alternative, we only The Notion of Utility know that this gain belongs to the interval [ v, v ]. Group Decision . . . • The overall gain u + v can thus take any value from We Must Take . . . the interval Home Page def = { u + v : u ∈ [ u, u ] , v ∈ [ v, v ] } . [ u, u ] + [ v, v ] Title Page ◭◭ ◮◮ • It is easy to check that ◭ ◮ [ u, u ] + [ v, v ] = [ u + v, u + v ] . Page 11 of 72 • Thus, the additivity requirement about the fair prices Go Back takes the form Full Screen P ([ u + v, u + v ]) = P ([ u, u ]) + P ([ v, v ]) . Close Quit