Abstract
Operator monotone functions play an important role in economics. We show that 2-monotonicity is equivalent to decreasing relative risk premium, a notion recently introduced in microeconomics. We also show that an operator monotone function is risk vulnerable, a notion introduced by Gollier and Pratt.
Highlights
Daniel Bernoulli [1] gave in 1738 an explanation of the Saint Petersburg paradox stated in 1713 by his cousin, Nicolas Bernoulli, in a letter to Pierre Raymond de Montmort
We show that 2-monotonicity is equivalent to decreasing relative risk premium, a notion recently introduced in microeconomics
We show that an operator monotone function is risk vulnerable, a notion introduced by Gollier and Pratt
Summary
Daniel Bernoulli [1] gave in 1738 an explanation of the Saint Petersburg paradox stated in 1713 by his cousin, Nicolas Bernoulli, in a letter to Pierre Raymond de Montmort. Probabilities may be subjective and formed by the decision maker as a result of his expectations of the unknown future In his seminal work on the foundation of statistics, Savage [7] gave an axiomatic description of preferences such that both the utility function (up to the composition with a strictly increasing affine transformation) and the subjective probabilities can be uniquely derived from the decision maker’s preferences, provided they satisfy a few natural conditions. There is empirical evidence for the claim that some decision makers may have preferences that cannot be represented by expected utility [2] This notion goes back to Knight [5], who introduced the distinction between risk and uncertainty, where risky events have assigned probabilities while others in Knight’s words are ‘ambiguous events for which ordinary probabilities cannot be defined’. In this paper we only deal with decision problems under risk and assume that decision makers are ‘probabilistic sophisticated’
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