Understanding ‘Ensemble-Average’ vs. 'Time-Average’

The TASK: Measure 10,000 coin flips

This could be a pain in ass. Who has time, right?

A quant will code once to measure the10,000 coin-flips.

This is an Ensemble Average.

However, Time-Average is where one person flips 10,000 times to measure the 10,000 coin-flips. This route is almost never studied because your quant likely got his PhD studying ergodic systems.

In ergodic systems, either method (time or ensemble averaging) yields the same result 99% of the time.

It’s ok to use either an ‘ensemble-average’ or a ‘time-average’ interchangeably because you will get the same result irrespective of your choice.

…but this is not so with non-ergodic systems like wealth dynamics.

Don’t let your quant interchange the concept of 'time’ vs ‘ensemble’ averages.  

He will try saying, “Markets bounce back”...or “markets mean-revert”.

Just respond with, “That’s nice, but if my trading account goes bankrupt, it will stay that way.”  

The 'time-average’ of your wealth can approach zero as time goes forward even though the ‘ensemble average’ of your historical wealth may be increasing.

Origins?

The word “ergodicity” was coined during the development of statistical mechanics. Ludwig Boltzmann, an Austrian physicist, invented it.

He called ergodic systems 'monodic’ at first.

A 'monodic’ system is one that possesses only one (mon) path (odos) through the space of its possible states.

The word became "ergodic" because Boltzmann considered systems of energy (or work = ergon), and the idea was that the one path covers all the states allowed by that energy (the energy shell).

Time-averages became interesting because that’s what he has always observed in physics

…but…

…time-averages take time to compute. Thus, the very clever insight of Boltzmann was that, under special conditions, “Hey, let’s just calculate the ‘ensemble average’ and it will coincide with the 'time-average’ (the one corresponding to reality).  

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Boltzmann’s short-cut to arriving at the required 'time-average’ by way of the ‘ensemble-average’ doesn’t work for finance

…because “Behavior-Over-Time” in no way connects to the idea of Ensemble-Average

Why Does This Stand?

…because it’s just not possible to compute time-averages for a system with 10^24 molecules bouncing around. He had to simplify the mathematics even if that meant resorting to a particularly horrendous short-cut. 

Scientists have long confirmed that, for most systems, it just doesn’t matter – but for finance or economics the situation is different.  

If you think about what matters to people, and what evolution has taught us, it’s pretty clear that it’s our performance over time that matters

…and not our average-performance over simultaneous parallel copies of ourselves.

Let’s say we roll a dice: “1”, you get shot, and if any other number appears, you live. Your quant will start with telling you that the mathematical expectation-value of this game is 3.5.

So according to the expectation value, you’ll live. Great game!

…but if you get shot rolling a “1” then nobody lives to see the next roll so as to meet the 3.5 average expectancy.

What matters is not the expectation-value of wealth increase, but the average-over-time of wealth increase.

The only reason smart people use ‘expectation values’ in physics is that they are the same as the ‘time-averages’ in those systems.

But ‘time-averages’ are what scientists are really after even though they don’t compute them.

All because it’s a pain-in-the-ass to wait for one person to do 10,000 trials over time when a computer can do 10,000 trials one time in an instant.