Need for Decision Making Need for Decision . . . How Decisions Under . . . Fair Price Approach: . . . How Much For an Interval? Case of Interval . . . a Set? a Twin Set? a p-Box? Case of Set-Valued . . . Case of Kaucher . . . A Kaucher Interval? Case of Triples Case of Twin Intervals Towards an Home Page Economics-Motivated Title Page Approach to Decision ◭◭ ◮◮ ◭ ◮ Making Under Uncertainty Page 1 of 25 Go Back Joe Lorkowski and Vladik Kreinovich University of Texas at El Paso, El Paso, TX 79968, USA Full Screen lorkowski@computer.org, vladik@utep.edu Close Quit
Need for Decision Making Need for Decision . . . 1. Need for Decision Making How Decisions Under . . . • In many practical situations: Fair Price Approach: . . . Case of Interval . . . – we have several alternatives, and Case of Set-Valued . . . – we need to select one of these alternatives. Case of Kaucher . . . • Examples: Case of Triples Case of Twin Intervals – 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 25 – a medical doctor needs to select a treatment for a Go Back patient. Full Screen Close Quit
Need for Decision Making Need for Decision . . . 2. Need for Decision Making Under Uncertainty How Decisions Under . . . • Decision making is easier if we know the exact conse- Fair Price Approach: . . . quences of each alternative selection. Case of Interval . . . Case of Set-Valued . . . • Often, however: Case of Kaucher . . . – we only have an incomplete information about con- Case of Triples sequences of different alternative, and Case of Twin Intervals – we need to select an alternative under this uncer- Home Page tainty. Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 25 Go Back Full Screen Close Quit
Need for Decision Making Need for Decision . . . 3. How Decisions Under Uncertainty Are Made How Decisions Under . . . Now Fair Price Approach: . . . • Traditional decision making assumes that: Case of Interval . . . Case of Set-Valued . . . – for each alternative a , Case of Kaucher . . . – we know the probability p i ( a ) of different outcomes i . Case of Triples • It can be proven that: Case of Twin Intervals Home Page – preferences of a rational decision maker can be de- scribed by utilities u i so that Title Page – an alternative a is better if its expected utility ◭◭ ◮◮ def = � u ( a ) p i ( a ) · u i is larger. ◭ ◮ i Page 4 of 25 Go Back Full Screen Close Quit
Need for Decision Making Need for Decision . . . 4. Hurwicz Optimism-Pessimism Criterion How Decisions Under . . . • Often, we do not know these probabilities p i . Fair Price Approach: . . . Case of Interval . . . • For example, sometimes: Case of Set-Valued . . . • we only know the range [ u, u ] of possible utility Case of Kaucher . . . values, but Case of Triples • we do not know the probability of different values Case of Twin Intervals within this range. Home Page • It has been shown that in this case, we should select Title Page an alternative s.t. α H · u + (1 − α H ) · u → max. ◭◭ ◮◮ • Here, α H ∈ [0 , 1] described the optimism level of a ◭ ◮ decision maker: Page 5 of 25 • α H = 1 means optimism; Go Back • α H = 0 means pessimism; Full Screen • 0 < α H < 1 combines optimism and pessimism. Close Quit
Need for Decision Making Need for Decision . . . 5. Fair Price Approach: An Idea How Decisions Under . . . • 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- Case of Set-Valued . . . uation Case of Kaucher . . . – by declaring a price that we are willing to pay to Case of Triples get involved in this situation. Case of Twin Intervals • 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 6 of 25 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 Making Need for Decision . . . 6. Case of Interval Uncertainty How Decisions Under . . . • Ideal case: we know the exact gain u of selecting an Fair Price Approach: . . . alternative. Case of Interval . . . Case of Set-Valued . . . • A more realistic case: we only know the lower bound Case of Kaucher . . . u and the upper bound u on this gain. Case of Triples • Comment: we do not know which values u ∈ [ u, u ] are Case of Twin Intervals 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 7 of 25 • Since we know that u , we have u ≤ P ([ u, u ]). Go Back Full Screen Close Quit
Need for Decision Making Need for Decision . . . 7. Case of Interval Uncertainty: Monotonicity How Decisions Under . . . • Case 1: we keep the lower endpoint u intact but in- Fair Price Approach: . . . crease the upper bound. Case of Interval . . . Case of Set-Valued . . . • This means that we: Case of Kaucher . . . – keeping all the previous possibilities, but Case of Triples – we allow new possibilities, with a higher gain. Case of Twin Intervals 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 8 of 25 • 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 Making Need for Decision . . . 8. Additivity: Idea How Decisions Under . . . • Let us consider the situation when we have two conse- Fair Price Approach: . . . quent independent decisions. Case of Interval . . . Case of Set-Valued . . . • We can consider two decision processes separately. Case of Kaucher . . . • We can also consider a single decision process in which Case of Triples we select a pair of alternatives: Case of Twin Intervals 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 9 of 25 – 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 Making Need for Decision . . . 9. Additivity: Case of Interval Uncertainty How Decisions Under . . . • 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 . . . Case of Set-Valued . . . • About the gain v from the second alternative, we only Case of Kaucher . . . know that this gain belongs to the interval [ v, v ]. Case of Triples • The overall gain u + v can thus take any value from Case of Twin Intervals the interval Home Page def [ u, u ] + [ v, v ] = { u + v : u ∈ [ u, u ] , v ∈ [ v, v ] } . Title Page ◭◭ ◮◮ • It is easy to check that ◭ ◮ [ u, u ] + [ v, v ] = [ u + v, u + v ] . Page 10 of 25 • 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
Need for Decision Making Need for Decision . . . 10. Fair Price Under Interval Uncertainty How Decisions Under . . . • By a fair price under interval uncertainty , we mean a Fair Price Approach: . . . function P ([ u, u ]) for which: Case of Interval . . . Case of Set-Valued . . . • u ≤ P ([ u, u ]) ≤ u for all u and u Case of Kaucher . . . ( conservativeness ); Case of Triples • if u = v and u < v , then P ([ u, u ]) ≤ P ([ v, v ]) Case of Twin Intervals ( monotonicity ); Home Page • ( additivity ) for all u , u , v , and v , we have Title Page P ([ u + v, u + v ]) = P ([ u, u ]) + P ([ v, v ]) . ◭◭ ◮◮ • Theorem: Each fair price under interval uncertainty ◭ ◮ has the form Page 11 of 25 P ([ u, u ]) = α H · u + (1 − α H ) · u for some α H ∈ [0 , 1] . Go Back • Comment: we thus get a new justification of Hurwicz Full Screen optimism-pessimism criterion. Close Quit
Recommend
More recommend
