[m0rc]

I remember the first time I heard someone (N. Taleb) expressing a negative opinion about economists because they were not "getting" the concept of ergodicity (in addition to other issues not relevant for this discussion). Initially, I didn't get the idea either. At some later point in time, I read Kelly's and Gellman's papers [1][2] with some effort, and although I was able to follow the arguments I didn't found anything surprising. That is, I didn't get the idea nor the implications.

It was during the reading of O. Peters and A. Adamou "Ergodicity Economics" [3] that I better understood the idea.

Imagine a basic gamble that repeats indefinitely. In each iteration and with equal probability, the player can win 60% more or lose 40% of the initial capital. For this gamble, the ensemble average (aka expectation) of the player's wealth one step ahead is simply 1/2 x 160% + 1/2 x 60% = 110%. A good gamble, right? However, the time average of the same step (i.e. the average gain of a single individual playing) is sqrt((1.6)x(0.6)) x 100 = 98%. So, the individual looses money with time. This was quite surprising to me although obvious a posteriori given that the multiplicative stochastic process is not ergodic. In other words, this simple gamble shows that the expectation does not have the intuitive meaning we sometimes assign to it specially for some repetitive gambles.

In other words. the time and ensemble average differ in general for non-ergodic stochastic processes and in particular for multiplicative stochastic processes (note that for additive processes the expectation of the wealth increment can be used).

And here comes the important implication... Given that several economic processes can be modeled as a first approximation as multiplicative random processes (e.g. stock markets, real investments, GDP growth, etc.), it is not a rational strategy to use the ensemble average (aka expectation of wealth increment) to take some economic decisions.

There are several implications of the above simple fact including the optimality of the Kelly criterion; the optimal leverage being below 1 in all cases involving multiplicative processes; the incorrect measurement of inequality; or the known inadequacy of the average income, instead of the median, to measure the average well-being to name a few.

A possible controversial corollary of the above is that the concept of utility is unnecessary and incorrect as a first approximation to the micro-economic behaviour. Instead an ergodic measurable should be used. In the specific cases of multiplicative stochastic processes the difference of the walth logarithm is ergodic and a rational decision maker should use it to optimize his wealth. This will require further debate within the scientific community because it is not clear that what is an optimal decision is a good model for the people's behavior. In any case, if the expected utility is not optimal, it also does not make much sense as a model for the Homo Economicus.

In any case, I really recommend reading instead of rushing to conclusions [3].

[1] J. L. Kelly, A new interpretation of information rate. Bell System Technical Journal, 35 (1956), 917-926. [2] O. Peters and M. Gell-Mann. Evaluating gambles using dynamics. Chaos, 26:23103, February 2016. [3] Peters, Ole, and Alexander Adamou. "Ergodicity economics." London Mathematical Laboratory (2018).

[m0rc]

>the optimal leverage being below 1 in all cases involving multiplicative processes

This is wrong.

Instead I should have written: "there is an optimal leverage point, most likely close to 1. The optimal leverage point does not depend on the individual risk preferences of the investor".