Let’s step back, and take a completely fresh look at the problem.

First, we consider financial decisions without uncertainty, which is very similar to the original idea of discounting. In the second step, we generalize by introducing noise. Placing considerations of time and ergodicity centre stage, we will arrive at a clear interpretation both of discounting and of utility theory, without appealing to subjective psychology or indeed other forms of personalization.

Financial decisions without uncertainty

A gamble without uncertainty is just a payment. A trivial model would be: we accept positive payments and reject negative ones. But what if we have to choose between two payments, or payment streams, at different times?

In this case, one consideration must be some form of a growth rate. For instance, I may choose between a job that offers $12,000 per year, and another that offers $2,000 per month. Let’s say the jobs are identical in all other respects: I would then choose the one that pays $2,000 per month — not because $2,000 is the greater payment (it isn’t), but because the payments correspond to a higher (additive) growth rate of my wealth. I would maximize

$$g_{\mathrm{a}} = \frac{\Delta x}{\Delta t}$$ (3)

Alternatively, I may have a choice between two savings accounts. One pays 4% per year, the other 1% per month — again, it’s the growth rate I would optimize: in this case the exponential growth rate

$$g_{\mathrm{e}} = \frac{{\Delta \ln x}}{\Delta t}$$ (4)

Since ∆t divides a difference in a generally nonlinear function of wealth, time now enters with a clear meaning but in potentially quite complicated ways — linearly (called hyperbolic in economics) as in equation (3) or exponentially as in equation (4).

Additive earnings and multiplicative returns on investments are the two most common processes that change our wealth, but we could think of other growth processes whose growth rates would have different functional forms. For example, the growth rate for the sigmoidal growth curves, in biology, of body mass versus time, has a different functional form10. For an arbitrary growth process x(t), the general growth rate is

$$g = \frac{{\Delta v\left( x \right)}}{\Delta t}$$ (5)

where v(x) is a monotonically increasing function chosen such that g does not change in time. Additive and multiplicative growth correspond to v a (x) = x and v e (x) = ln x. Generalizing, v(x) is the inverse of the process x(t) at unit rate, denoted

$$v\left( x \right) = x_1^{\left( { - 1} \right)}\left( x \right)$$ (6)

For financial processes, fitting more general functions often results in an interpolation between linear and logarithmic, maybe in a square-root function, or a similar small tweak.

Ergodic observables

Real-life financial decisions usually come with a degree of uncertainty. We let the model reflect this by introducing noise. But how?

To perturb the process in a consistent way, we remind ourselves that what’s constant about the process in the absence of noise is the growth rate. If we perturb that with a constant-amplitude noise, the scale of the perturbation will be time independent in v-space, and in that sense adapted to the dynamics. That’s easily done by writing equation (5) in differential form, replacing the function g by its (constant) value, γ, say, rearranging and adding the noise (here represented by a Wiener term dW with amplitude σ)

$${\mathrm{d}}v = \gamma {\mathrm{d}}t + \sigma {\mathrm{d}}W$$ (7)

The process itself is found by integrating equation (7) and solving for x. For our two key examples, this produces Brownian motion (with v a = x) and geometric Brownian motion (with v e = ln x).

The growth rates for these processes are no longer constant because they are noisy. But the lack of constancy is due to nothing other than the noise. Using the nomenclature introduced in equation (1), the relevant growth rates are ergodic observables of their respective processes. By design, their (time or ensemble) averages tell us what tends to happen over time.

This is not the case for wealth itself, and it exposes the expected wealth model as physically naive. The expected wealth change simply does not reflect what happens over time (unless the wealth dynamic is additive; Fig. 2). The initial correction — expected utility theory — overlooked the physical problem and jumped to psychological arguments, which are hard to constrain and often circular.

Fig. 2: Randomly generated trajectories of the repeated example gamble. The example gamble is given in equation (2). The expectation value, 〈x〉, (blue line) is the average over an infinite ensemble, but it doesn’t reflect what happens over time. The ergodic growth rate for the process (slope of the red line) tells us what happens to a typical individual trajectory. 150 trajectories are shown, each consists of 1,000 repetitions. Full size image

Growth rate optimization is now sometimes called ‘ergodicity economics’. This doesn’t mean that ergodicity is assumed — quite the opposite: it refers to doing economics by asking explicitly whether something is ergodic, which is often not the case. As we have seen, ergodicity economics is a perspective that arises from constructing ergodic observables for non-ergodic (growth) processes.

Mapping

Both expected utility theory and ergodicity economics introduce nonlinear transformations of wealth, and the equations that appear in the two frameworks can be very similar. More precisely, the mapping is this: the appropriate growth rate for a given process is formally identical to the rate of change of a specific utility function

$$g = \frac{{\Delta v\left( x \right)}}{\Delta t} = \frac{{\Delta u\left( x \right)}}{\Delta t}$$ (8)

The time average of this growth rate is identical to the rate of change of the specific expected utility function — because of ergodicity.

Despite the mapping, conceptually the two approaches couldn’t be more different, and ergodicity economics stays closer to physical reality.

Expected utility theory is a loose end of the mapping because the only constraints on the utility function it provides are loose references to psychology. While some view this as a way to ensure generality, my second criticism is more severe and I’m unable to resolve it: in maximizing the expectation value — an ensemble average over all possible outcomes of the gamble — expected utility theory implicitly assumes that individuals can interact with copies of themselves, effectively in parallel universes (the other members of the ensemble). An expectation value of a non-ergodic observable physically corresponds to pooling and sharing among many entities. That may reflect what happens in a specially designed large collective, but it doesn’t reflect the situation of an individual decision-maker.

Expected utility theory computes what happens to a loosely specified model of my psychology averaged across a multiverse. But I do not live spread out across a multiverse, let alone harvest the average psychological consequences of the actions of my multiverse clones.

Ergodicity economics, in contrast, computes what will happen to my physical wealth as time goes by, without appeal to an intangible psychology or a multiverse. We all live through time and suffer the physical consequences (and psychological consequences, for that matter) of the actions of our younger selves.

With ergodicity economics, the psychological insight that some people are systematically more cautious than others attains a physical interpretation. Perhaps people aren’t so different, but their circumstances are. Someone maximizing growth in an additive process would appear to be brave: v a = u brave , whereas the same person doing the same thing but for a multiplicative process would appear to be scared: v e = u scared . Note also the scale dependence of these statements: the same ∆x (in dollars) corresponds to large logarithmic wealth changes for a poor person and to small logarithmic changes for a rich person — the latter are linearizable, and the rich person looks brave.

It also makes historical sense: in the early days of probability theory there was a firm belief that things should be expressed in terms of expectation values. For that to make any sense in the context of individuals making financial decisions, an ergodic observable had to be created. Expected utility theory — unknowingly, because ergodicity hadn’t been invented — did just that. But because of the lack of conceptual clarity, the entire field of economics drifted in a direction that places too much emphasis on psychology.