Ergodicity economics

Ergodicity economics questions whether expected value is a useful indicator of performance over time. In doing so it builds on existing critiques of the use of expected value in the modeling of economic decisions. Such critiques started soon after the introduction of expected value in 1654. For instance, expected-utility theory was proposed in 1738 by Daniel Bernoulli ^[2] as a way of modeling behavior which is inconsistent with expected-value maximization. In 1956, John Kelly devised the Kelly criterion by optimizing the use of available information, and Leo Breiman later noted that this is equivalent to optimizing time-average performance, as opposed to expected value.^[3]

The ergodicity economics research programme originates in two papers by Ole Peters in 2011, a theoretical physicist and current external professor at the Santa Fe Institute.^[4] The first studied the problem of optimal leverage in finance and how this may be achieved by considering the non-ergodic properties of geometric brownian motion.^[5] The second paper applied principles of non-ergodicity to propose a possible solution for the St. Petersburg paradox.^[6] More recent work has suggested possible solutions for the equity premium puzzle, the insurance puzzle, gamble-selection, probability weighting, and has provided insights into the dynamics of income inequality.^[7]

The concept of ergodicity and non-ergodicity in economic processes can be illustrated with a repeated multiplicative coin toss, an instance of the binomial multiplicative process.^[8] It demonstrates how an expected-value analysis can indicate that a gamble is favorable although the gambler is guaranteed to lose over time.

Multiplicative Coin Toss

Definition

In this thought experiment, discussed in,^[7] a person participates in a simple game where they toss a fair coin.

If the coin lands heads, the person gains 50% on their current wealth; if it lands tails, the person loses 40%.

The game shows the difference between the expected value of an investment, or bet, and the time-average or real-world outcome of repeatedly engaging in that bet over time.

Calculation of Expected Value

Denoting current wealth by ${\displaystyle x(t)}$ , and the time when the payout is received by ${\displaystyle t+\delta t}$ , we find that wealth after one round is given by the random variable ${\displaystyle x(t+\delta t)}$ , which takes the values ${\displaystyle 1.5\times x(t)}$ (for heads) and ${\displaystyle 0.6\times x(t)}$ (for tails), each with probability ${\displaystyle p_{\text{H}}=p_{\text{T}}=1/2}$ . The expected value of the gambler's wealth after one round is therefore

${\displaystyle {\begin{aligned}E[x(t+\delta t)]&=p_{\text{H}}\times 1.5x(t)+p_{\text{T}}\times 0.6x(t)\\&=1.05x(t).\end{aligned}}}$

By induction, after ${\displaystyle T}$ rounds expected wealth is ${\displaystyle E[x(t+T\delta t)]=1.05^{T}x(t)}$ , increasing exponentially at 5% per round in the game.

This calculation shows that the game is favorable in expectation -- its expected value increases with each round played.

Calculation of Time-Average

The time-average performance indicates what happens to the wealth of a single gambler who plays repeatedly, reinvesting their entire wealth every round. Due to compounding, after ${\displaystyle T}$ rounds the wealth will be

${\displaystyle x(t+T\delta t)=\prod _{\tau =1}^{T}r_{\tau }x(t),}$

where we have written ${\displaystyle r_{\tau }}$ to denote the realized random factor by which wealth is multiplied in the ${\displaystyle \tau ^{\text{th}}}$ round of the game (either ${\displaystyle r_{\tau }=r_{\text{H}}=1.5}$ for heads; or ${\displaystyle r_{\tau }=r_{\text{T}}=0.6}$ , for tails). Averaged over time, wealth has grown per round by a factor

${\displaystyle {\bar {r}}_{T}=\left({\frac {x(t+T\delta t)}{x(t)}}\right)^{1/T}.}$

Introducing the notation ${\displaystyle n_{\text{H}}}$ for the number of heads in a sequence of coin tosses we re-write this as

${\displaystyle {\bar {r}}_{T}=\left(r_{\text{H}}^{n_{\text{H}}}r_{\text{T}}^{T-n_{\text{H}}}\right)^{1/T}=r_{\text{H}}^{n_{\text{H}}/T}r_{\text{T}}^{(T-n_{\text{H}})/T}.}$

For any finite ${\displaystyle T}$ , the time-average per-round growth factor, ${\displaystyle {\bar {r}}_{T}}$ , is a random variable. The long-time limit, found by letting the number of rounds diverge ${\displaystyle T\to \infty }$ , provides a characteristic scalar which can be compared with the per-round growth factor of the expected value. The proportion of heads tossed then converges to the probability of heads (namely 1/2), and the time-average growth factor is

${\displaystyle \lim _{T\to \infty }{\bar {r}}_{T}=\left(r_{\text{H}}r_{\text{T}}\right)^{\frac {1}{2}}\approx 0.95.}$

Discussion

The comparison between expected value and time-average performance illustrates an effect of broken ergodicity: over time, with probability one, wealth decreases by about 5% per round, in contrast to the increase by 5% per round of the expected value.

In December 2020, Bloomberg news published an article titled "Everything We’ve Learned About Modern Economic Theory Is Wrong"^[9] discussing the implications of ergodicity in economics following the publication of a review of the subject in Nature Physics.^[7] Morningstar covered the story to discuss the investment case for stock diversification.^[10]

In the book Skin in the Game, Nassim Nicholas Taleb suggests that the ergodicity problem requires a rethinking of how economists use probabilities.^[11] A summary of the arguments was published by Taleb in a Medium article in August 2017.^[12]

In the book The End of Theory, Richard Bookstaber lists non-ergodicity as one of four characteristics of our economy that are part of financial crises, that conventional economics fails to adequately account for, and that any model of such crises needs to take adequate account of.^[13] The other three are: computational irreducibility, emergent phenomena, and radical uncertainty.^{[citation needed]}

In the book The Ergodic Investor and Entrepreneur, Boyd and Reardon tackle the practical implications of non-ergodic capital growth for investors and entrepreneurs, especially for those with a sustainability, circular economy, net positive, or regenerative focus.^[14]

James White and Victor Haghani discuss the field of ergodicity economics in their book The Missing Billionaires.^[15]

The approach and relevance of the ergodicity economics research program has been criticised significantly by mainstream economists. They argue that the program misstates the content and predictions of mainstream economic theory in criticizing it, and that the basic ergodicity economics model makes obviously false predictions about behavior.^[16] An experiment^[17] carried out by neuroscientists in Denmark which "would corroborate ergodicity economics and falsify expected utility theory" has also been particularly criticised for its methods and for overstating its results.^[18]

• Santa Fe Institute
• St. Petersburg paradox