Tuesday, December 3, 2019

"The ergodicity problem in economics"

Double Hat Tips up front:
From Nature Physics, December 2, 2019:
Abstract

The ergodic hypothesis is a key analytical device of equilibrium statistical mechanics. It underlies the assumption that the time average and the expectation value of an observable are the same. Where it is valid, dynamical descriptions can often be replaced with much simpler probabilistic ones — time is essentially eliminated from the models. The conditions for validity are restrictive, even more so for non-equilibrium systems. Economics typically deals with systems far from equilibrium — specifically with models of growth. It may therefore come as a surprise to learn that the prevailing formulations of economic theory — expected utility theory and its descendants — make an indiscriminate assumption of ergodicity. This is largely because foundational concepts to do with risk and randomness originated in seventeenth-century economics, predating by some 200 years the concept of ergodicity, which arose in nineteenth-century physics. In this Perspective, I argue that by carefully addressing the question of ergodicity, many puzzles besetting the current economic formalism are resolved in a natural and empirically testable way.

Main

Ergodic theory is a forbiddingly technical branch of mathematics. Luckily, for the purpose of this discussion, we will need virtually none of the technicalities. We will call an observable ergodic if its time average equals its expectation value, that is, if it satisfies Birkhoff’s equation


(1)
Here, f is determined by the system’s state ω. On the left-hand side, the state in turn depends on time t. On the right-hand side, a timeless P(ω) assigns weights to ω. If equation (1) holds we can avoid integrating over time (up to the divergent averaging time, T, on the left), and instead integrate over the space of all states, Ω (on the right). In our case P(ω) is given as the distribution of a stochastic process. In systems with transient behaviour, that may require defining P(ω) as the t → ∞ limit of a time dependent density function P(ω; t).

Famously, ergodicity is assumed in equilibrium statistical mechanics, which successfully describes the thermodynamic behaviour of gases. However, in a wider context, many observables don’t satisfy equation (1). And it turns out a surprising reframing of economic theory follows directly from asking the core ergodicity question: is the time average of an observable equal to its expectation value?
At a crucial place in the foundations of economics, it is assumed that the answer is always yes — a pernicious error. To make economic decisions, I often want to know how fast my personal fortune grows under different scenarios. This requires determining what happens over time in some model of wealth. But by wrongly assuming ergodicity, wealth is often replaced with its expectation value before growth is computed. Because wealth is not ergodic, nonsensical predictions arise. After all, the expectation value effectively averages over an ensemble of copies of myself that cannot be accessed.
This key error is patched up with psychological arguments about human behaviour. The consequences are numerous, but over the centuries their root cause has become invisible in the growing formalism. Observed behaviour deviates starkly from model predictions. Paired with a firm belief in its models, this has led to a narrative of human irrationality in large parts of economics. Scientifically, this deserves some reflection: the models were exonerated by declaring the object of study irrational.

I stumbled on this error about a decade ago, and with my collaborators at the London Mathematical Laboratory and the Santa Fe Institute I have identified a number of long-standing puzzles or paradoxes in economics that derive from it. If we pay close attention to the ergodicity problem, natural solutions emerge. We therefore have reason to be optimistic about the future of economic theory....
....MUCH MORE
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