[Paper Review] The time interpretation of expected utility theory

Ole Peters, Alexander Adamou|arXiv (Cornell University)|Jan 11, 2018

Complex Systems and Time Series Analysis13 references3 citations

TL;DR

This paper establishes a formal correspondence between ergodicity economics and expected utility theory by showing that the ergodicity transformation in ergodicity economics corresponds to the utility function in expected utility theory. It generalizes growth optimality beyond additive and multiplicative dynamics, demonstrating that risk preferences are determined by wealth dynamics, and provides a physical, time-based foundation for utility functions, resolving long-standing conceptual issues in decision theory.

ABSTRACT

Ergodicity economics is a new branch of economic theory that notes the conceptual difference between time averages and expectation values, which coincide only for ergodic observables. It postulates that individual agents maximise the time average growth rate of wealth, known widely as growth optimality. This contrasts with the dominant behavioural model in economics, expected utility theory, in which agents maximise expectation values of changes in psychologically transformed wealth. Historically, growth optimality was explored for additive and multiplicative gambles. Here we apply it to a general class of wealth dynamics, extending the range of economic situations where it may be used. Moreover, we show a correspondence between growth optimality and expected utility theory, in which the ergodicity transformation in the former is identified as the utility function in the latter. This correspondence offers a theoretical basis for choosing utility functions and predicts that wealth dynamics are strong determinants of risk preferences.

Motivation & Objective

• To resolve conceptual inconsistencies in expected utility theory by grounding it in time-averaged growth rates rather than expectation values.
• To generalize growth optimality beyond additive and multiplicative wealth dynamics to a broad class of stochastic wealth processes.
• To establish a formal correspondence between ergodicity transformations in ergodicity economics and utility functions in expected utility theory.
• To provide a mechanistic, physically meaningful basis for utility functions, eliminating the arbitrariness of psychological assumptions.
• To challenge the long-standing belief that utility functions must be bounded, showing unbounded functions are mathematically preferable and physically more natural.

Proposed method

• Derives the time-average growth rate of wealth under general stochastic dynamics, using ergodicity transformations to extract a constant growth rate from non-ergodic processes.
• Applies the ergodicity transformation to map wealth dynamics into a process where time averages correspond to expectation values, enabling direct comparison with expected utility theory.
• Uses stochastic differential equations to model wealth dynamics, with the ergodicity transformation derived from the Fokker-Planck equation and the resulting stationary distribution.
• Shows that maximizing the time-average growth rate under a given dynamic is mathematically equivalent to maximizing the expected utility of a transformed variable, where the transformation is the inverse of the ergodicity map.
• Derives the wealth distribution from the dynamic by transforming the normally distributed utility process via the inverse ergodicity transformation.
• Demonstrates that unbounded utility functions are necessary for consistent time-averaging and avoids finite-time singularities or time-boundedness.

Experimental results

Research questions

• RQ1 How can ergodicity economics be extended beyond additive and multiplicative wealth dynamics to general stochastic processes?
• RQ2 What is the formal mathematical correspondence between ergodicity transformations in ergodicity economics and utility functions in expected utility theory?
• RQ3 Can risk preferences be derived from the underlying dynamics of wealth rather than assumed via arbitrary utility functions?
• RQ4 Why is the boundedness of utility functions a problematic assumption, and what are the consequences of assuming unboundedness?
• RQ5 How can the wealth distribution be analytically derived from a given dynamic using the ergodicity transformation?

Key findings

• The ergodicity transformation in ergodicity economics is mathematically equivalent to the utility function in expected utility theory, providing a physical interpretation for utility.
• For any stochastic wealth dynamic, maximizing time-average growth rate is equivalent to maximizing expected utility of a transformed variable, with the transformation derived from the dynamic.
• Risk preferences are not arbitrary but are fully determined by the underlying wealth dynamics, explaining why individuals behave as if they have specific utility functions.
• Wealth distributions can be analytically derived from the dynamic using the inverse ergodicity transformation, resulting in heavy-tailed distributions when fluctuations diminish with wealth.
• Unbounded utility functions are not only permissible but necessary for consistent time-averaging; boundedness leads to unphysical finite-time singularities or time-boundedness.
• The model resolves the conceptual flaw in expected utility theory by replacing expectation values with time-averaged growth rates, restoring empirical and mechanistic relevance.