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Discounting axioms imply risk neutrality

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Abstract

Although most applications of discounting occur in risky settings, the best-known axiomatic justifications are deterministic. This paper provides an axiomatic rationale for discounting in a stochastic framework. Consider a representation of time and risk preferences with a binary relation on a real vector space of vector-valued discrete-time stochastic processes on a probability space. Four axioms imply that there are unique discount factors such that preferences among stochastic processes correspond to preferences among present value random vectors. The familiar axioms are weak ordering, continuity and nontriviality. The fourth axiom, decomposition, is non-standard and key. These axioms and the converse of decomposition are assumed in previous axiomatic justifications for discounting with nonlinear intraperiod utility functions in deterministic frameworks. Thus, the results here provide the weakest known sufficient conditions for discounting in deterministic or stochastic settings. In addition to the four axioms, if there exists a von Neumann-Morgenstern utility function corresponding to the binary relation, then that function is risk neutral (i.e., affine). In this sense, discounting axioms imply risk neutrality.

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Acknowledgements

I was fortunate to have been a student at Columbia University. In Columbia College my advisor was George E. Kimball, an operations research pioneer for whom the Kimball Medal of INFORMS is named, and a prominent theoretical chemist. The mathematics teachers who influenced me were Samuel Eilenberg, Bernard O. Koopman (after whom the Koopman Prize of the Military Applications section of INFORMS is named), and Serge Lange. Cyrus Derman was my icon of an academician then and he remained one for more than fifty years.

Even to a twenty-year old student it was obvious that Professor Derman’s papers would have enduring importance. His interests were broad, he was intense but fair-minded, he had a passion for chamber music (he could have attended Curtis Institute as a violinist), and he was an enthusiastic tennis player and skier. In his courses I first encountered linear programming, stochastic processes, and his student and graduate assistant, Arthur F. Veinott, Jr., who was a major influence on me a few years later at Stanford University. Professor Derman encouraged my fascination with Markov decision processes, he supervised my master’s essay, and more than fifty years later I still benefit from his tutelage.

I valued Professor Robert Rosenthal’s friendship highly, and his penetrating questions (Rosenthal 1987) led to Sect. 5. He died in 2002.

I am grateful to Professor James C. Alexander for comments and the argument on which the proof of Lemma 3(a) is based, to Professors Eugene Feinberg and Charu Sinha for comments, and to an anonymous commentator on a precursor of this paper who noted that confining Y to 0 in (A3) permitted a counterexample.

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  1. Department of Operations, Weatherhead School of Management, Case Western Reserve University, 10600 Euclid Ave., Cleveland, OH, 44106-7235, USA

    Matthew J. Sobel

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Correspondence to Matthew J. Sobel.

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In memory of Professors Cyrus Derman and Robert Rosenthal.

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Sobel, M.J. Discounting axioms imply risk neutrality. Ann Oper Res 208, 417–432 (2013). https://doi.org/10.1007/s10479-012-1066-9

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