This is part two of a running thought process around decision making amidst uncertainty. The key concept this month is Ergodicity. This is a $10 word to keep in your back pocket for those
times when you want to ensure a quick change in conversation partner at a cocktail party.
Whether a system is ergodic or non-ergodic dictates whether a person can expect to achieve the statistically average result. Imagine we're playing a game of coin toss and every time my coin shows tails I will collect $100. A flip of heads gets nothing. If I
flip a coin 100 times, I can reasonably expect to eventually accumulate fifty heads and fifty tails as long as it’s a fair coin. Over time, I should collect $5,000 as long as I complete all 100 flips. Easy money. Additionally, we'll also let 100 people each flip a coin one time (still 100 flips total). Every person who gets a result of tails will also get the $100. In this case, we also expect fifty of those tosses to be heads and fifty to be tails – again assuming they’re all fair coins. $5,000
will be paid out.
The above scenario is considered ergodic because one person can expect their individual results over 100 flips to mirror the aggregate results from the 100 people each flipping one time. The same should hold if both flip 1,000 times or 10,000 times. The individual outcome can be expected to match the population outcome as long
as both complete the full series of flips. Now imagine another scenario.
Let’s say we are in a hotel with 100 rooms. We randomly select fifty of the rooms to put $100 inside and the other fifty each will contain a hungry tiger. Just like in the coin flip game, the participants get to keep the money when they open a room with cash inside.
We’ll assign 100 different people one door to open each. Afterward, we’ll reset and I will try to open all 100 doors.
Question: Is my expected return in the tiger scenario the same as it was in the coin toss scenario? Afterall, the outcome of the population was the same in both. Half of them got the $100 and half did not. The return was
the same for the population in both scenarios - $5,000 was collected. Shouldn't I expect to walk away from both with $5,000?
Answer: Of course not.
In the coin flip situation, I was able to
complete all 100 tosses even though I probably would have had some streaks of tails (wins) and streaks of heads (losses) in a row. I could expect my individual result of fifty heads, fifty tails and $5,000 to mirror the results achieved by the larger group (the population outcome) over the long run despite streaky results throughout. In the tiger situation, the group of 100 is also going to have fifty people with $100 each (wins) and fifty people who don't get paid (losses). It’s the same
aggregate outcome as the coin flip for the population statistically. It will not be the same for me though.
The difference for me as an individual is there’s no guarantee I’ll get to open all 100 doors. I might be $300 richer after opening the first three rooms (a streak of wins) but then get chomped by a tiger on the fourth one and that’s the
end of my results. When that happens, I don’t get to open the 96 other doors. The losses literally ate up all the remaining potential wins. My actual odds of getting the $5,000 are far below 1% because I would have to be able only to open the doors with cash inside and never open a room with a tiger. A perfect 50-win streak.
This is
referred to as the “Game Over” problem. As an individual, I should not expect to achieve the population's average result if there's a chance I won't get to keep playing the game. This makes the tiger scenario non-ergodic and should change the way I evaluate the potential risks and rewards involved. There’s obviously hyperbole at work in this example but the concept applies to any non-ergodic situation where a game over is a possibility.
Here's the problem: Many situations in life are assumed to be ergodic when they really aren't. We first need to assess whether it is reasonable to expect to achieve the average rate of return based on the facts of our specific situation and what might cause us to have a game over outcome. For example, financial plans are often built on expectations of achieving an average rate of return. For
an individual investor, whether they earn the 30-year average rate of return largely depends on whether they can afford to stay fully invested long enough to have their results mirror the population results (e.g. whether they get to open all 100 doors).
Understanding ergodicity should cause us to manage risk in a such a way that we reduce
the likelihood of a game over result when possible. Doing so increases our odds of achieving the average return and reaching our goals. We can do this by building margins of safety to our strategy that let us survive undesirable outcomes and accumulate enough results to move our individual returns toward the long-term average over time.
How this concept can play out in planning decisions and portfolio design will be the third part of the discussion next month.
Thank you for your trust and your time.
