Ergodicity vs Expected Value: Why Smart Bets Can Still Ruin You

One of the most dangerous misconceptions in decision-making is that maximizing expected value always leads to optimal outcomes. This belief has led countless investors, entrepreneurs, and professionals to financial ruin despite making "mathematically correct" decisions.

The missing piece? Ergodicity.

The coin flip that ruins you

Consider this bet: flip a coin. Heads, you gain 50% of your wealth. Tails, you lose 40%.

The expected value is positive:

(0.5 × 50%) + (0.5 × -40%) = +5% per flip

Traditional economics says: take this bet every time. The expected value is positive!

But let's see what actually happens. With $100, after one heads and one tails (in any order):

• $100 → $150 (heads) → $90 (tails), or
• $100 → $60 (tails) → $90 (heads)

Either way, you end up with $90 - a 10% loss.

After 10 flips, your expected wealth taking into account the +5% per flip average is about $163. But what happens in practice, is that you're more likely to find yourself with a loss.

This is not a paradox: it's the difference between what happens to the average of a population versus what happens to an individual over time.

The Russian Roulette problem

To further understand this phenomenon, let's consider the gambler's game of Russian Roulette. The player takes a gun, empties the cylinder, and puts back a single bullet. Then, he spins the cylinder to randomize the position of the bullet. Finally, he takes the gun to his head. After staring at death for a few seconds, he pulls the trigger. If he survives, he collects a prize, usually in the tens of thousands of dollars. (Obviously, do not try this at home, or anywhere else.)

If the prize of winning one round of Russian Roulette is $10,000, its "expected value" is:

(5/6 × $10,000) + (1/6 × $0) = $8,333

What if you play it 10 times? The average outcome is not 10 times the average returns of playing it once, but death.

That's because your probabilities of survival decrease with each round played.

This reveals the fundamental flaw in expected value thinking: it assumes you can somehow experience the average across all possible outcomes. But you only live one life, experiencing one timeline.

In ergodic systems, time averages and ensemble averages converge. In non-ergodic systems, they diverge, often dramatically.

As the joke goes: "5 in 6 economists think Russian Roulette is a great investment."ShareCopy

The key difference: irreversibility

What makes a system non-ergodic? Irreversibility.

When losses are irreversible, losing a bet doesn't just mean losing that bet but also all future ones (and thus, missing their returns).

Most important decisions in life are non-ergodic:

• Investing: Not only losing 100% means game over, but losing $200 on a $500 investment means losing not just those $200 but also all future returns these $200 could have generated.
• Career: Some behaviors mean you lose not just the current job but all future ones (if they make you unhireable)
• Health: Certain injuries cannot be fully recovered from
• Relationships: Trust, once broken, may never fully recover

Why "risk aversion" is rational

There is a common belief that people are irrationally risk averse. It's the result of experiments such as the following:

"Here is a game. You flip a coin. If it's heads, I give you $1000. If it's tails, you give me $950. Do you want to participate?"

From a naive point of view, the expected return of playing the bet is $1000 times 50% (the chances of winning) minus $950 times 50% (the chances of losing). That would be $500 - $475 = $25. On average, every time you play the game, you're expected to win $25. This makes the gamble apparently desirable. And yet, if researchers go around asking the question to real people, most decline. This led behavioral economists to conclude that people are irrationally risk-averse.

Are they, though?

If people had infinite cash, they could play the game as long as they wanted. The law of large numbers would kick in, their lifetime outcome would converge to their expected outcome, and they would realize the expected win of $25 per coin flipped.

However, real people do not have infinite cash. They can only play this game a few times before emptying their bank accounts or having to quit the game. Some cannot even afford to lose once.

For real people, the limitation on the number of times they can play can transform their lifetime outcome of a gamble into negative.

An example

Imagine that you have a sum of $1000 in your pocket.

After one iteration of the game:

• You might have won the toss and won $1000
• You might have lost the toss and lost $950

The average is a win of $25, as expected. However, if you are offered to play a second time, you can only afford to play if you won the first toss. Therefore, you can expect to win $25 from the second toss only if you won the first one.

This means that in four parallel universes:

• In the first, you won both tosses, and you're up by $2000
• In the second, you won the first toss and lost the second one. You're up $50
• In the third, you lost the first toss. You're down $950
• In the fourth, you also lost the first toss and cannot play again. You're down $950

After two iterations of the game, you have won an average of just:

($2000 + $50 - $950 - $950) / 4 = $37.5

This is surprising! If you had infinite wealth, you would have won an average of $25 per bet times two equals $50. But because your wealth is finite, your average win is lower: only $37.5.

More importantly, you have a one-in-four chance of winning a lot of money, one-in-four chance of winning a modest amount, and one-in-two chance of losing a significant amount.

The behavioral economists who called people "irrationally risk-averse" are the irrational ones. Declining a positive expected value bet when you have finite resources isn't a bias; it's wisdom.ShareCopy

Further readings: Ole Peters's and Alexander Adamou's papers discuss this problem and contain additional examples of how (non-)ergodicity explains the hidden rationality of some risk aversion and of other behaviors that would be irrational in an ideal ergodic world. As far as I know, he was the first to propose ergodicity as the solution to many otherwise puzzling behaviors.

Practical implications

For investors

• Don't bet the farm on "positive expected value" opportunities
• Use position sizing that prevents catastrophic losses
• Diversification is not about sacrificing returns for survival, but about using survival to maximize long-term returns.

For entrepreneurs

• Don't risk everything on a single venture
• Build runway before taking major risks
• Have a fallback plan for irreversible failures

For life decisions

• Consider the worst-case scenario, not just the average
• Distinguish between recoverable and unrecoverable mistakes
• Instead of optimizing for expected outcomes, maximize the distribution of outcomes

The two questions to always ask

Before making any risky decision, ask yourself:

1. Can I recover from the worst outcome? If not, the expected value is irrelevant.
2. Am I confusing population outcomes with individual outcomes? Just because some succeed doesn't mean you will.
3. Am I optimizing the best possible outcome, or the likely outcome?

Learn more

Understanding the difference between ergodicity and expected value is one of the most valuable mental models for long-term success. If you want to go deeper:

• Read my book on ergodicity for practical frameworks and more examples
• Explore ergodicity economics for the academic foundations
• See how this applies to investing decisions

Survival is necessary for optimization. Expected value is a useful tool for decisions you'll repeat thousands of times with small stakes, but for everything else, think about the outcome of you taking that decision multiple times, not about the average outcome of many people taking that decision once.ShareCopy