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How to Estimate Hurwicz Value of the Union of Two Intervals If We Know the Hurwicz Values for Both Intervals?

Mahdokht Afravi, Gerardo Cervantes, Xavier Martinez, Matthew Melvin, Victor Vargas, Ana Zepeda, and Vladik Kreinovich

Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA, mafravi@miners.utep.edu, gcervantes8@miners.utep.edu, xamartinez2@miners.utep.edu, mjmelvin@miners.utep.edu, vevargascor@miners.utep.edu, alzepeda@miners.utep.edu, vladik@utep.edu

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1. Decision Making Under Interval Uncertainty

• The decision maker’s preferences can be described by

the corresponding utility function: – the larger its value, – the better the alternative.

• In practice, we often do not know the exact values of

the utility corresponding to each alternative.

• 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].

• In such situations, decision theory recommends select-

ing the alternative with the largest value of H(u) = α · u + (1 − α) · u.

• This expression is known as Hurwicz value.

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2. Decision Under Interval Uncertainty (cont-d)

• Reminder: H(u) = α · u + (1 − α) · u.
• The parameter α describes the optimism level of the

decision maker; e.g.: – the value α = 1 corresponds to perfect optimism, while – the value α = 0 corresponds to perfect pessimism.

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3. Question

• In some cases, an action can result in two possible out-

comes.

• For each outcome, we know the utility interval u1 and u2.
• We do not know which outcome will happen.
• So, overall, the set of possible utility values is the union

u1 ∪ u2 of these intervals.

• To be precise, in general, it is the interval hull

[min(u1, u2), max(u1, u2)] of this union.

• If we know Hurwicz values of the two intervals, what

can we say about the Hurwicz value of their union?

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4. Our Result

• Our result is that

min(H(u1), H(u2)) ≤ H(u1∪u2) ≤ max(H(u1), H(u2)).

• Without losing generality, we will prove only the right

inequality.

• We consider two possible cases:

– the case when the order between the bounds in the same for both bounds, i.e., ∗ either u1 ≤ u2 and u1 ≤ u2, ∗ or u2 ≤ u1 and u2 ≤ u1; – and the case when the order between lower bounds is different from the order between upper bounds.

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5. Our Result (cont-d)

• In the first case, without losing generality, we can as-

sume that u1 ≤ u2 and u1 ≤ u2.

• Then, the union has the form [u1, u2].
• Here, H(u2) = α · u2 + (1 − α) · u2 and H(u1 ∪ u2) =

α · u2 + (1 − α) · u1.

• Since u1 ≤ u2, we conclude that H(u1 ∪ u2) ≤ H(u2)

and thus, H(u1 ∪ u2) ≤ max(H(u1), H(u2)).

• In the second case, without losing generality, we can

assume that u1 ≤ u2 and u2 ≤ u1.

• In this case, min(u1, u2) = u1 and max(u1, u2) = u1.
• Thus u1 ∪u2 = u1 hence H(u1 ∪u2) = H(u1) and thus

too, H(u1 ∪ u2) ≤ max(H(u1), H(u2)).

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6. References

• L. Hurwicz, Optimality Criteria for Decision Making

Under Ignorance, Cowles Commission Discussion Pa- per, Statistics, No. 370, 1951.

• V. Kreinovich, “Decision making under interval un-

certainty (and beyond)”, In: P. Guo and W. Pedrycz (eds.), Human-Centric Decision-Making Models for So- 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.