Abstract
A random variable can be equivalently regarded to as a function or as a “set”, namely, that of the points lying below (when positive) and above (when negative) its graph. The second approach, proposed by Segal in 1989, is known as the measure (or measurement) representation approach. On a technical ground, it allows for using Measure theory tools instead of Functional analysis ones, thus making often possible to reach new and deeper conclusions. On an interpretative ground, it makes clear how expectation and expected utility, either classical or à la Choquet, are structurally analogous and, moreover, it allows for dealing with new and more general types of expectations including, e.g., state dependence.
Starting from the measurement approach and from a decision-theoretical result by Castagnoli and LiCalzi (2006), we present a new representation theorem in the same perspective and, finally, we propose a definition of generalised expectations and two different concepts of associativity that can be imposed to them.
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References
Amarante, M.: Foundations of neo-bayesian statistics. J. Econ. Theory 144, 2146–2173 (2009)
Beccacece, F., Cillo, A.: Applying the benchmarking procedure: a decision criterion of choice under risk. Theory Dec. 61, 75–91 (2006)
Bordley, R.F.: Foundations of Target-based Decision Theory. In: Brachinger, H.W., Monney, P. (eds.) Decision Analysis, from Encyclopedia of Life Support Systems (EOLSS). Eolss Publishers, Oxford (2002), http://www.eolss.net
Bordley, R.F., LiCalzi, M.: Decision Analysis Using Targets instead of Utility Functions. Decis. Econ. Finance 23, 53–74 (2000)
Castagnoli, E.: Qualche riflessione sull’utilità attesa. Ratio Math. 1, 51–59 (1990)
Castagnoli, E., LiCalzi, M.: Expected Utility without Utility: Constant Risk Attitude. Rend. com. studi econ. XXX/XXXI, 145–160 (1993)
Castagnoli, E., LiCalzi, M.: Expected Utility without Utility: A Model of Portfolio Selection. In: D’Ecclesia, R.L., Zenios, S.A. (eds.) Operations Research Models in Quantitative Finance, pp. 95–111. Physica-Verlag, Heidelberg (1994)
Castagnoli, E., LiCalzi, M.: Expected Utility without Utility. Theory Dec. 41, 281–301 (1996)
Castagnoli, E., LiCalzi, M.: Benchmarking real-valued acts. Games Econ. Behav. 57, 236–253 (2006)
Castagnoli, E., Favero, G., Maccheroni, F.: Marchetto, il prezzatore perfetto (Elogio dell’internalità). Edizioni Čakuntala, Milano (still in progress) (2009)
Cigola, M., Modesti, P.: On SSB Utility Theory. In: Bertocchi, M., Cavalli, E., Komlosi, S. (eds.) Modelling Techniques for Financial Markets and Bank Management, pp. 272–284. Physica-Verlag, Heidelberg (1996)
Debreu, G.: Topological methods in cardinal utility theory. In: Arrow, K.J., Karlin, S., Suppes, P. (eds.) Mathematical Methods in the Social Sciences, 1959: Proceedings of the First Stanford Symposium. Stanford Mathematical Studies in the Social Sciences, vol. 4, pp. 16–26. Stanford University Press, Stanford (1960) (reprinted as Cowles Foundation paper 156)
Greco, G.H.: Integrale monotono. Rend. Sem. Mat. Univ. Padova 57, 149–166 (1977)
Klement, E.P., Mesiar, R., Pap, E.: A Universal Integral as Common Frame for Choquet and Sugeno integrals. Work in progress (2009)
Modesti, P.: Lottery-dependent Utility via Stochastic Benchmarking. Theory Dec. 55, 45–57 (2003)
Nehring, K.: Capacities and probabilistic beliefs: a precarious coexistence. Math. Soc. Sci. 38, 197–213 (1999)
Savage, L.J.: The Foundations of Statistics. Wiley, New York (1951) (revised edition, Dover, New York (1972))
Segal, U.: Anticipated Utility: A Measure Representation Approach. Ann. Oper. Res. 19, 359–373 (1989)
Segal, U.: The Measure Representation: A Correction. J. Risk Uncertainty 6, 99–107 (1993)
Simon, H.A.: A Behavioral Model of Rational Choice. Quart. J. Econ. 69, 99–118 (1955)
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Castagnoli, E., Favero, G. (2010). From Benchmarks to Generalised Expectations. In: Greco, S., Marques Pereira, R.A., Squillante, M., Yager, R.R., Kacprzyk, J. (eds) Preferences and Decisions. Studies in Fuzziness and Soft Computing, vol 257. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15976-3_3
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DOI: https://doi.org/10.1007/978-3-642-15976-3_3
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