Bounded, Sigmoid Utility for Insurance Applications

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-m...

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Bibliographic Details
Published inAsia-Pacific journal of risk and insurance Vol. 11; no. 1; pp. 6 - 24
Main Authors Gao, Siwei, Powers, Michael R.
Format Journal Article
LanguageEnglish
Published Singapore De Gruyter 01.01.2017
Walter de Gruyter GmbH
Subjects
Bids
bounded utility
Decision theory
dominance reasoning
Expected values
heavy-tailed gains/losses
Insurance applications
Reinsurance
sigmoid utility
St. Petersburg paradox
two-envelope paradox
Utility functions
von Neumann-Morgenstern utility
Willingness to pay
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Summary: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.
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ISSN:1793-2157
2153-3792
DOI:10.1515/apjri-2016-0009