Bounded, Sigmoid Utility for Insurance Applications

Gao Siwei () and Powers Michael R. ()
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Gao Siwei: Department of Accounting, Finance and Information Systems, Eastern Kentucky University, 521 Lancaster Ave., Richmond, KY 40475, United States of America
Powers Michael R.: Finance Department, Tsinghua University, 386C Weilun Building, Beijing, Beijing 100084, China

Asia-Pacific Journal of Risk and Insurance, 2017, vol. 11, issue 1, 19

Abstract: Applying a well-known argument of Karl Menger to an insurance version of the St. Petersburg Paradox (in which the decision maker is confronted with losses, rather than gains), one can assert that von Neumann-Morgenstern utility functions are necessarily concave upward and bounded below as decision-maker wealth tends to negative infinity. However, this argument is subject to two potential criticisms: (1) infinite-mean losses do not exist in the real world; and (2) the St. Petersburg Paradox derives its force from empirical observation (i. e., that actual decision makers would not agree to an arbitrarily large insurance bid price to transfer an infinite-mean loss), and thus does not impart logical necessity. In the present article, these two criticisms are addressed in turn. We first show that, although infinite-mean insurance losses technically do not exist, they do provide a reasonable model for certain large (i. e., excess and reinsurance) property-liability indemnities. We then employ the Two-Envelope Paradox to demonstrate the logical necessity of concave-upward, lower-bounded utility for arbitrarily small (i. e., negative) values of wealth. Finally, we note that recognizing the bounded, sigmoid nature of utility functions challenges certain fundamental understandings in the economics of insurance demand, and can lead to vastly different conclusions regarding the bid price for insurance.

Keywords: von Neumann-Morgenstern utility; St. Petersburg paradox; heavy-tailed gains/losses; two-envelope paradox; dominance reasoning; sigmoid utility; bounded utility (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1515/apjri-2016-0009

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