# Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems

## Abstract

**:**

## 1. Introduction

## 2. The Classical Two-Envelope Paradox

#### 2.1. Naive Form

#### 2.2. Enhanced Form

#### 2.3. Continuous Payoffs

**Proposition 1:**For any continuous increasing utility function and distribution parameters α and θ, dominance reasoning is preserved if and only if the recurrence inequality

#### 2.4. Discrete Payoffs

**Proposition 2:**For any continuous increasing utility function and distribution parameters m and p, dominance reasoning is preserved if and only if the recurrence inequality

## 3. The St. Petersburg Two-Envelope Paradox

#### 3.1. Conditional Independence

#### 3.2. Continuous Payoffs

**Proposition 3:**For any continuous increasing utility function and distribution parameters α and θ, dominance reasoning is preserved if and only if

#### 3.3. Discrete Payoffs

**Proposition 4:**For any continuous increasing utility function and distribution parameters m and p, dominance reasoning is preserved if and only if

## 4. Conclusions

## Acknowledgments

## Appendix

## Proof of Proposition 1:

## Proof of Proposition 2:

## Proof of Proposition 3:

## Proof of Proposition 4:

## Conflicts of Interest

## References

- B. Nalebuff. “The Other Person’s Envelope Is Always Greener.” J. Econ. Perspect. 3, 1 (1989): 171–181. [Google Scholar] [CrossRef]
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Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems. *Risks* **2015**, *3*, 26-34.
https://doi.org/10.3390/risks3010026

**AMA Style**

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Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems. *Risks*. 2015; 3(1):26-34.
https://doi.org/10.3390/risks3010026

**Chicago/Turabian Style**

Powers, Michael R.
2015. "Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems" *Risks* 3, no. 1: 26-34.
https://doi.org/10.3390/risks3010026