# Decision Making Under Interval Uncertainty (and Beyond)

## Abstract

To make a decision, we must find out the user’s preference, and help the user select an alternative which is the best—according to these preferences. Traditional utility-based decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is often unable to select one of these alternatives. In this chapter, we show how we can extend the utility-based decision theory to such realistic (interval) cases.

### Keywords

Decision Making Interval Uncertainty Utility## Notes

### Acknowledgments

This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grant 1 T36 GM078000-01 from the National Institutes of Health, and by a grant on F-transforms from the Office of Naval Research. The authors is greatly thankful to the anonymous referees for valuable suggestions.

### References

- 1.Aliev, R., Huseynov, O., Kreinovich, V.: Decision making under interval and fuzzy uncertainty: towards an operational approach. In: Proceedings of the Tenth International Conference on Application of Fuzzy Systems and Soft Computing ICAFS’2012, Lisbon, Portugal, 29–30 August 2012Google Scholar
- 2.Aubinet, M., Vesala, T., Papale, D. (eds.): Eddy Covariance - A Practical Guide to Measurement and Data Analysis. Springer, Dordrecht (2012)Google Scholar
- 3.Binder, A.B., Roberts, D. L.: Criteria for Lunar Site Selection, Report No. P-30, NASA Appollo Lunar Exploration Office and Illinois Institute of Technology Research Institute, Chicago, Illinois, January 1970Google Scholar
- 4.Cobb, C.W., Douglas, P.H.: A theory of production. Am. Econ. Rev.
**18**(Supplement), 139–165 (1928)Google Scholar - 5.de Finetti, B.: Theory of Probability, Wiley, New York, Vol. 1, 1974; Vol. 2, 1975Google Scholar
- 6.Ehrgott, M., Gandibleux, X. (eds.): Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys. Springer, New York (2002)Google Scholar
- 7.Ferson, S., Kreinovich, V., Hajagos, J., Oberkampf, W., Ginzburg, L.: Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty, Sandia National Laboratories (2007), Publ. 2007–0939Google Scholar
- 8.Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1969)Google Scholar
- 9.Fishburn, P.C.: Nonlinear Preference and Utility Theory. Johns Hopkins University Press, Baltimore (1988)MATHGoogle Scholar
- 10.Fountoulis, I., Mariokalos, D., Spyridonos, E., Andreakis, E.: Geological criteria and methodology for landfill sites selection. In: Proceedings of the 8th International Conference on Environmental Science and Technology, Greece, 8–10 September 2003, pp. 200–207Google Scholar
- 11.Gardeñes, E., et al.: Modal intervals. Reliable Comput.
**7**, 77–111 (2001)CrossRefMATHGoogle Scholar - 12.Giang, P.H.: Decision with dempster-shafer belief functions: decision under ignorance and sequential consistency. Int. J. Approximate Reasoning
**53**, 38–53 (2012)MathSciNetCrossRefMATHGoogle Scholar - 13.Guo, P., Tanaka, H.: Decision making with interval probabilities. Eur. J. Oper. Res.
**203**, 444–454 (2010)MathSciNetCrossRefMATHGoogle Scholar - 14.Guo, P., Wang, Y.: Eliciting dual interval probabilities from interval comparison matrices. Inf. Sci.
**190**, 17–26 (2012)CrossRefMATHGoogle Scholar - 15.Helversen B.V., Mata R.: Losing a dime with a satisfied mind: positive affect predicts less search in sequential decision making, Psychology and Aging,
**27**, 825–839 (2012)Google Scholar - 16.Hurwicz, L.: Optimality Criteria for Decision Making Under Ignorance, Cowles Commission Discussion Paper, Statistics, No. 370, 1951Google Scholar
- 17.Jaimes A., Salayandia L., Gallegos I.: New cyber infrastructure for studying land-atmosphere interactions using eddy covariance techniques. In: Abstracts of the 2010 Fall Meeting American Geophysical Union AGU’2010, San Francisco, California, 12–18 December 2010Google Scholar
- 18.Jaimes, A., Herrera, J., GonzÍez, L., Ramìrez, G., Laney, C., Browning, D., Peters, D., Litvak, M., Tweedie, C.: Towards a multiscale approach to link climate, NEE and optical properties from a flux tower, robotic tram system that measures hyperspectral reflectance, phenocams, phenostations and a sensor network in desert a shrubland. In: Proceedings of the FLUXNET and Remote Sensing Open Workshop: Towards Upscaling Flux Information from Towers to the Globe FLUXNET-SpecNet’2011, Berkeley, California, 7–9 June 2011Google Scholar
- 19.Jaimes, A.: A cyber-Tool to Optimize Site Selection for Establishing an Eddy Covraince and Robotic Tram System at the Jornada Experimental Range. University of Texas, El Paso (2008)Google Scholar
- 20.Jaimes, A., Tweedie, C., Kreinovich, V., Ceberio, M.: Scale-invariant approach to multi-criterion optimization under uncertainty, with applications to optimal sensor placement, in particular, to sensor placement in environmental research. Int. J. Reliab. Saf.
**6**(1–3), 188–203 (2012)Google Scholar - 21.Kiekintveld, C., Kreinovich, V.: Efficient approximation for security games with interval uncertainty, In: Proceedings of the AAAI Spring Symposium on Game Theory for Security, Sustainability, and Health GTSSH’2012, Stanford, 26–28 March 2012Google Scholar
- 22.Kintisch, E.: Loss of carbon observatory highlights gaps in data. Science
**323**, 1276–1277 (2009)CrossRefGoogle Scholar - 23.Kreinovich, V.: Decision making under interval uncertainty, In: Abstracts of the 15th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Verified Numerical Computation SCAN’2012. Novosibirsk, Russia, September 2012Google Scholar
- 24.Kreinovich, V., Starks, S.A., Iourinski, D., Kosheleva, O., Finkelstein, A.: Open-ended configurations of radio telescopes: a geometrical analysis. Geombinatorics
**13**(2), 79–85 (2003)MathSciNetMATHGoogle Scholar - 25.Kyburg, H.E., Smokler, H.E. (eds.): Studies in Subjective Probability. Wiley, New York (1964)Google Scholar
- 26.Luce R.D., Krantz D.H., Suppes P., Tversky A.: Foundations of measurement, vol. 3, Representation, Axiomatization, and Invariance, Academic Press, California (1990)Google Scholar
- 27.Luce, R.D., Raiffa, R.: Games and Decisions: Introduction and Critical Survey. Dover, New York (1989)MATHGoogle Scholar
- 28.Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Pennsylvania (2009)CrossRefMATHGoogle Scholar
- 29.Myerson, R.B.: Game theory. Analysis of conflict, Harvard University Press, Cambridge (1991)Google Scholar
- 30.Nash, J.F.: The bargaining problem. Econometrica
**28**, 155–162 (1950)MathSciNetCrossRefGoogle Scholar - 31.Nash, J.: Two-person cooperative games. Econometrica
**21**, 128–140 (1953)MathSciNetCrossRefMATHGoogle Scholar - 32.Nguyen, H.T., Kreinovich, V., Wu, B., Xiang G.: Computing Statistics under Interval and Fuzzy Uncertainty, Springer, Berlin (2012)Google Scholar
- 33.Nguyen, H.T., Kreinovich, V.: Applications of Continuous Mathematics to Computer Science. Kluwer, Dordrecht (1997)MATHGoogle Scholar
- 34.Pfanzangl, J.: Theory of Measurement. Wiley, New York (1968)Google Scholar
- 35.Raiffa, H.: Decision Analysis. McGraw-Hill, Columbus (1997)Google Scholar
- 36.Roth, A.: Axiomatic Models of Bargaining. Springer, Berlin (1979)CrossRefMATHGoogle Scholar
- 37.Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954)MATHGoogle Scholar
- 38.Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, Orlando (1985)MATHGoogle Scholar
- 39.Shary, S.P.: A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Comput.
**8**, 321–418 (2002)MathSciNetCrossRefMATHGoogle Scholar - 40.Steuer, R.E.: Multiple Criteria Optimization: Theory, Computations, and Application. Wiley, New York (1986)Google Scholar
- 41.Stevens, S.S.: On the theory of scales of measurement. Science
**103**, 677–680 (1946)CrossRefMATHGoogle Scholar - 42.Troffaes, M.C.M.: Decision making under uncertainty using imprecise probabilities. Int. J. Approximate Reasoning
**45**, 7–29 (2007)MathSciNetCrossRefGoogle Scholar - 43.Walsh, C.: Monetary Theory and Policy. MIT Press, Cambridge (2003)Google Scholar