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Monotone additive statistics

Xiaosheng Mu, Luciano Pomatto, Philipp Strack, Omer Tamuz

Abstract

The expectation is an example of a descriptive statistic that is monotone with respect to stochastic dominance, and additive for sums of independent random variables. We provide a complete characterization of such statistics, and explore a number of applications to models of individual and group decision-making. These include a representation of stationary monotone time preferences, extending the work of Fishburn and Rubinstein (1982) to time lotteries. This extension offers a new perspective on risk attitudes toward time, as well as on the aggregation of multiple discount factors. We also offer a novel class of nonexpected utility preferences over gambles which satisfy invariance to background risk as well as betweenness, but are versatile enough to capture mixed risk attitudes.

Monotone additive statistics

Abstract

The expectation is an example of a descriptive statistic that is monotone with respect to stochastic dominance, and additive for sums of independent random variables. We provide a complete characterization of such statistics, and explore a number of applications to models of individual and group decision-making. These include a representation of stationary monotone time preferences, extending the work of Fishburn and Rubinstein (1982) to time lotteries. This extension offers a new perspective on risk attitudes toward time, as well as on the aggregation of multiple discount factors. We also offer a novel class of nonexpected utility preferences over gambles which satisfy invariance to background risk as well as betweenness, but are versatile enough to capture mixed risk attitudes.

Paper Structure

This paper contains 40 sections, 32 theorems, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Monotone Additive Statistics
  4. Monotone Stationary Time Preferences
  5. Domain and Axioms
  6. Representation
  7. Implications for Risk Attitudes toward Time
  8. Stationarity, Time Invariance and Dynamic Consistency
  9. Aggregation of Preferences over Time Lotteries
  10. Preferences Over Gambles
  11. Background-risk Invariant Preferences
  12. Risk Aversion
  13. Mixed Risk Aversion
  14. Comparative Risk Attitudes
  15. Betweenness
  16. ...and 25 more sections

Key Result

Theorem 1

$\Phi \colon L^\infty \to \mathbb{R}$ is a monotone additive statistic if and only if there exists a (unique) Borel probability measure $\mu$ on $\overline{\mathbb{R}}$ such that for every $X \in L^\infty$

Figures (2)

  • Figure 1: The c.d.f.s of $X$ (blue) and $Y$ (orange).
  • Figure 2: The c.d.f.s of $X+Z$ (blue) and $Y+Z$ (orange).

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition
  • Definition
  • Definition
  • Claim 1
  • Definition
  • Proposition 1
  • Proposition 2
  • ...and 45 more