# Approximations of One-dimensional Expected Utility Integral of Alternatives Described with Linearly-Interpolated **p-**Boxes

**p-**

## Abstract

In the process of quantitative decision making, the bounded rationality of real individuals leads to elicitation of interval estimates of probabilities and utilities. This fact is in contrast to some of the axioms of rational choice, hence the decision analysis under bounded rationality is called fuzzy-rational decision analysis. Fuzzy-rationality in probabilities leads to the construction of *x*-ribbon and *p*-ribbon distribution functions. This interpretation of uncertainty prohibits the application of expected utility unless ribbon functions were approximated by classical ones. This task is handled using decision criteria *Q* under strict uncertainty—Wald, maximax, Hurwicz\({}_{\alpha }\), Laplace—which are based on the pessimism-optimism attitude of the decision maker. This chapter discusses the case when the ribbon functions are linearly interpolated on the elicited interval nodes. Then the approximation of those functions using a *Q* criterion is put into algorithms. It is demonstrated how the approximation is linked to the rationale of each *Q* criterion, which in three of the cases is linked to the utilities of the prizes. The numerical example demonstrates the ideas of each *Q* criterion in the approximation of ribbon functions and in calculating the *Q*-expected utility of the lottery.

### Keywords

Quantitative decision making Rational choice Interval estimates Bounded rationality Fuzzy-rationality Decision criteria under strict uncertainty Expected utility### References

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