Disclaimer: This is an ongoing story and I am not privy to any facts beyond what I read on the internet like everyone else, note this is all alleged activity and everything said here is strictly for entertainment and informational purposes.

There are a lot of topics that seem academic and semantic but that have really big real-world implications. One of those is the St. Petersburg Paradox and how it relates to my hobby horse topic of ergodicity.

This sounds boring but it managed to lose people ~$42 billion dollars as of the latest accounting. I am referring, of course, to the blow-up of crypto exchange FTX and the implosion of its founder, Sam Bankman-Fried’s (SBF).

If you have not been following it closely, the gist of what we know about the story so far is that there was an offshore crypto exchange (FTX) that was valued at ~$32 billion and had ~$10 billion in customer deposits but suddenly wasn’t able to process withdrawals and filed for Chapter 11 bankruptcy.

Here’s the thing about losing $42 billion. It’s pretty hard to do. You know, sometimes I pay too much for the extra guac too. But, losing $42 billion requires some serious miscalculations.

The SBF/FTX example is an important one because I believe many investors (and more economists) make a similar mistake in how they think about investing a bet sizing.

The good news is, it’s actually really easy to avoid if you understand what’s going on!

I first heard this thinking from Sam Bankman-Fried (SBF) in his interview with Tyler Cowen in March, 2022 where they had the following exchange:

COWEN: Should a Benthamite be risk-neutral with regard to social welfare?

BANKMAN-FRIED: Yes, that I feel very strongly about.

COWEN: Okay, but let’s say there’s a game: 51 percent, you double the Earth out somewhere else; 49 percent, it all disappears. Would you play that game? And would you keep on playing that, double or nothing?

BANKMAN-FRIED: With one caveat. Let me give the caveat first, just to be a party pooper, which is, I’m assuming these are noninteracting universes. Is that right? Because to the extent they’re in the same universe, then maybe duplicating doesn’t actually double the value because maybe they would have colonized the other one anyway, eventually.

COWEN: But holding all that constant, you’re actually getting two Earths, but you’re risking a 49 percent chance of it all disappearing.

BANKMAN-FRIED: Again, I feel compelled to say caveats here, like, “How do you really know that’s what’s happening?” Blah, blah, blah, whatever. But that aside, take the pure hypothetical.

COWEN: Then you keep on playing the game. So, what’s the chance we’re left with anything? Don’t I just St. Petersburg paradox you into nonexistence?

BANKMAN-FRIED: Well, not necessarily. Maybe you St. Petersburg paradox into an enormously valuable existence. That’s the other option.

At the time, this seemed completely insane to me to the point that I just assumed I had misunderstood some assumption he was making. However, subsequent events seem to suggest that SBF actually took this notion seriously and it’s an incredibly valuable lesson to understand why this makes no sense. To do that, we need to talk about the St. Petersburg Paradox and what it implies about investing and portfolio construction.

(Editorial Note: For the purposes of this piece, I am setting aside the fact that there are many allegations of fraud in this instance and you don’t really need to resort to the St. Petersburg Paradox to explain why fraud is bad and how it loses people’s money. However, there are, in my experience, a lot of non-fraudulent people that make a similar miscalculation to what the leaders at FTX and Alameda Research seem to have made. The fraud part means they lost a lot of other people’s money, not just their own as a result.)

The Problem of the Points

The story of FTX’s implosion begins, of course, in 1654 when Pascal and Fermat come up with the concept of expected value to solve the “problem of the points.”¹

The problem of points is a thought experiment: imagine a game of dice where the players are betting on the highest score in rolling a dice three times in a row. But, let’s say their mom finishes the meatloaf after only two rolls so they have to abandon their dice game before it’s done. In this situation, how is the “pot”, the total wager, to be distributed among the players in a fair manner?

Pascal and Fermat agreed that the fair solution is to give to each player the expectation value of his winnings: all possible outcomes of the game are calculated with their probability multiplied by their outcome and each player paid accordingly

This procedure uses the idea of parallel universes. Instead of considering only the state of the universe as it is, or will be, an infinity of additional equally likely universes is imagined.

Setting aside the fact that “fair” is a moral concept, not a mathematical one, it’s reasonable to say that most people consider this a fair solution. If one player was very likely to win, they probably deserve most of the pot. This is the foundation of the useful notion of expected value.

What is the St. Petersburg Paradox?

In 1713, mathematician Nicolaus Bernoulli put forth a question about the notion of expected value that came to be known as the St. Petersburg Paradox.

Consider a lottery where the winning amount is determined by the consecutive number of tails flipped with the winning amount doubling after each flip.

• First flip: If heads, the lottery pays $1, and the game ends. If tails, the coin is tossed again
• Second Flip: If heads, the lottery pays $2, and the game ends. If tails, the coin is tossed again.
• Third Flip: If heads, the lottery pays $4, and the game ends. If tails, the coin is tossed again.
• Fourth Flip: If heads, the lottery pays $8, and the game ends. If tails, the coin is tossed again.
• Fifth Flip: If heads, the lottery pays $16, and the game ends. If tails, the coin is tossed again.

This goes on ad infinitum: On heads, the lottery pays out double of the prior round and the game ends. On tails, the coin is tossed again.

Now, How much would you pay to play this game?

Pick a number in your head (it’s more fun than if you just skip ahead)

…..

Based on expected value theory, a “rational” person should be willing to pay any price for a ticket in this lottery. After all, there is a small probability that the coin will land on tails so many times in a row that the payout is infinity. And anything times infinity is infinity.

In reality, however, people are rarely willing to pay more than $10.² This is the St. Petersburg Paradox: why are people only willing to pay $10 for a game which has an infinite expected value?

The Utility Theory Explanation of The St. Petersburg Paradox

Why were people only willing to pay $10 to play this game? Bernoulli came up with a psychological and behavioral explanation.

They proposed that the desirability or “utility” associated with a financial gain depends not only on the gain itself but also on the wealth of the person who is making this gain.

This makes intuitive sense.

To the person who has no money, $500 could make a big impact on their life.

To the person who has $1,000,000, an extra $500 doesn’t really change anything for them.

Instead of computing the expectation value of the winnings, they proposed to compute instead the expectation value of the gain in utility.

Bernoulli came to the conclusion that the reason people were only willing to pay $10 rather than infinity dollars is because the marginal dollar becomes worth less and less to them.³

Bernoulli (1738) suggested a logarithmic function which was based on the intuition that the increase in wealth should correspond to an increase in utility that is inversely proportional to the wealth a person already has.

To put it into chart form, a log function looks something like this:

Going from having $2,000 to having $4,000 doesn’t double someone’s utility. It increases it modestly, but a lot less than going from $0 to $2,000.

Bernoulli’s solution was that lottery players base their decisions not on the expected monetary gain but instead on the expected gain in usefulness, and thus the paradox is thus resolved.

This assumption is at the foundation of the way economics is taught today and pretty much any Econ 101 textbook will include an explanation of utility theory.

Enter Stage Right: Ergodicity Economics

There is another explanation that makes a lot more sense to me published by Ole Peters in 2011: The time resolution of the St. Petersburg paradox.

Peter cites the notion of ergodicity. Ergodicity differentiates between time averages and ensemble averages. In an ergodic system, the time and ensemble averages are the same. In a non-ergodic system, they are different.

Take a game of Russian Roulette as an example. If you survive, you win $1,000,000.

If 6 different players play Russian Roulette and you conducted an after that fact survey, 5 out every 6 Russian roulette players would recommend it as a very exciting and profitable game.

You might roll the dice and take $1,000,000 to play Russian Roulette one time (though I wouldn’t advise it). But there’s no amount of money that would make you play it 6 times in a row. You are guaranteed to lose!

Russian Roulette is non-ergodic – you don’t get the same outcome of 6 people playing once as you do if one person plays six times.

In practice, pretty much all situations we face in our lives are non-ergodic. We do not live 100 simultaneous lives, we live one life through time. Anything we do which contains the risk of ruin (or large losses) then must be avoided to maximize long-term wealth growth.

The two mental pictures – many parallel cooperating trajectories versus a single trajectory unfolding over a long period – are at odds.

The resolution of the St. Petersburg paradox according to ergodicity economics, rather than utility theory, makes two assumptions:

• Rejection of parallel universes: To the individual who decides whether to purchase a ticket in the lottery or play Russian Roulette or make an investment, it is irrelevant how he may fare in a parallel universe.
• Acceptance of continuation of time: What matters to the individual is whether he makes decisions under uncertain conditions in such a way as to accumulate wealth over time.

According to this logic, the reason that people are only willing to pay $10 is not because the utility of the marginal dollar decreases, but because they know (if only subconsciously) that they aren’t living in multiple universes, but one life that unfolds over time and so they shouldn’t bet too much even when the expected value is infinite.⁴

Where it gets messy is that regardless of whether you assume people are using a logarithmic utility function OR they are assuming non-ergodicity, you get about the same answer. ⁵ on this point.

The fact that you end up with about the same answer has lead some people to say that it doesn’t really matter which one you use – it’s a distinction without a difference.

However, let’s go back to Sam Bankman-Fried (SBF)’s interview with Tyler Cowen:

COWEN: Let’s say there’s a game: 51 percent, you double the Earth out somewhere else; 49 percent, it all disappears. Would you play that game? And would you keep on playing that, double or nothing?

BANKMAN-FRIED: With one caveat. Let me give the caveat first, just to be a party pooper, which is, I’m assuming these are non-interacting universes. Is that right? Because to the extent they’re in the same universe, then maybe duplicating doesn’t actually double the value because maybe they would have colonized the other one anyway, eventually.

COWEN: But holding all that constant, you’re actually getting two Earths, but you’re risking a 49 percent chance of it all disappearing.

BANKMAN-FRIED: Again, I feel compelled to say caveats here, like, “How do you really know that’s what’s happening?” Blah, blah, blah, whatever. But that aside, take the pure hypothetical.

COWEN: Then you keep on playing the game. So, what’s the chance we’re left with anything? Don’t I just St. Petersburg paradox you into nonexistence?

BANKMAN-FRIED: Well, not necessarily. Maybe you St. Petersburg paradox into an enormously valuable existence. That’s the other option.

If you believe that the correct way to think about the problem is in terms of utility then that means a different utility function leads to a different strategy.

SBF did a Twitter thread along similar lines back in 2020. He gives an example of a game similar to the St. Petersburg Paradox with the odds tilted in your favor.

His logical reasoning was based on the notion of Utility theory.

The logic is exactly the same as Bernoulli: people are betting according to their utility function. Bernoulli hypothesized that most people have a logarithmic utility function like we saw in the chart, but it could vary from person to person which SBF pointed out. He then proposed a slightly different bet that is more unlikely to work but has much higher upside.

The Kelly Criterion

“Kelly” here references the Kelly Criterion, a theory from J. L. Kelly at Bell Labs in 1956. It is a formula for bet sizing (or investment sizing) that leads to more wealth than any other strategy in the long run.

For all intents and purposes, using the Kelly criterion is the practical implementation of Peters’ ergodic theory.⁶

The Kelly Criterion optimizes for time average wealth, not ensemble average wealth. That is, it assumes markets are non-ergodic – that you are one individual playing over time not a collection of individuals playing in parallel universes.⁷

At its simplest level, Kelly uses the odds you are getting on the bet times the probability of being right to determine the appropriate Fraction of your Bankroll to Allocate.

If you plug in the odds that SBF used in his example – a bet with a 10,000 to 1 payoff but only a 10% chance of being correct, you can see that indeed, 10% is the Kelly optimal amount of your bankroll to bet.

You can play with a Kelly Criterion calculator here to get a feel for how it works but generally, the idea is that the better your odds of winning and the higher the payout then the more you bet, but you never bet so much that you can go bust.

It is an interesting coincidence that using a logarithmic utility function gets you to the same answer as a time average. However, this is a bad assumption because then someone can just say “I don’t have a logarithmic utility function so I should bet differently.”

And this is exactly what SBF argued:

His justification is saying, in essence, “my utility function is not a logarithmic function, but more of a linear function. Since twice as much money means I can save twice as many people from dying from malaria and the value of human life is constant, it makes sense to wager more.”

If you assume the utility theory explanation, this actually makes perfect sense! If you believe that the reason betting only 10% makes sense is because your utility function is logarithmic rather than it is “rational” to bet more if you have a near-linear utility function.

Using the ergodic solution to the St. Petersburg Paradox rather than the utility solution leads to the opposite conclusion.

If your goal is long term growth, it never makes sense to bet more than 100% Kelly.

Consider this simulated example of a bet using 100% Kelly, half (50%) Kelly, and twice (200%) Kelly.

You can see that in the long run, the Kelly betting system is the most profitable and twice Kelly actually the least profitable.

If you have future bets, then the most important thing is not to go bankrupt and Kelly makes sure you can’t go bankrupt. In games where you know the odds such as the ones SBF proposed, there is always a way to calculate the Kelly Criterion.

You can use those calculations to come up with a Kelly Curve.

Here, the Y-axis represents the geometric growth rate, the X-axis represents bet size, and the Kelly-optimal bet lies at the highest point on the curve. In every field of application, the general shape of the graph will be the same, though the optimal bet size will vary based on the odds and chance of winning.

Source: Nick Yoder on the Kelly Criterion

Here’s the critical thing in this image: betting more than Kelly decreases your long-term growth rate.

It is not about being more risk tolerant because you have a different utility function!

Matt Hollerbach from Breaking the Market has a great recap of his discussion with SBF on this topic. Here’s the key point:

And now we see SBF has a very different view of the Kelly Criteria, and of what a Geometric growth rate is.

SBF thinks the all in bet maximizes the long term growth rate. pic.twitter.com/XIFUIuEq17

— Matt Hollerbach (@breakingthemark) November 11, 2022

SBF believed that the “all in” bet maximized long term growth because he was thinking about through the lens of utility rather than ergodicity or Kelly.

The Ricky Bobby Criterion

In the great philosophical treatise that is Talladega Nights, Ricky Bobby (Will Ferrel) repeats the slogan “If you ain’t first, you’re last.”

Desperate to win at any cost, Bobby exceeds his limitations (in our analogy, he bets more than 100% Kelly) and crashes his car. His declining performance subsequently gets him fired, his wife divorces him and he falls into a deep depression.

In trying to be first in the short term, at any cost, he ends up in last.

Caroline Ellison, the CEO of FTX’s sister company/proprietary hedge fund Alameda Research apparently had a Tumblr account where she uses effectively the same logic SBF did in his interview.

This is the Ricky Bobby Criterion: If you ain’t first, you’re last. If you aren’t maximizing your expected value in the context of a single bet, you are “lame.”

The Ricky Bobby Criterion suggests that you should always be maximizing your leverage and doubling down because if you aren’t always in first place, you might as well be in last place.

This is, of course, fucking insane. It’s not about going as fast as possible all the time, it’s about winning in the long run.

In Formula 1 racing, the most important thing on the car is actually the brakes! The better the braking ability, the faster you can go between the sharp turns. A long race with multiple laps around a circuit is like a non-ergodic path of compounding.

The objective is ending capital, how you fare at the end of the 40 laps. If you make a critical mistake at one of the sharp turns, the negative compounding effect means you aren’t winning the race (possibly not even finishing it). Better brakes doesn’t always matter in the short run, but it’s the thing that matters most in the long run. ⁸

Greater than Kelly bet sizing is to be avoided not because people are “risk-averse” or “lame” but because their lives are non-ergodic. We cannot go back in time or move to parallel universes.

Where Risk Tolerance is Relevant

Behavioral aspects and personal circumstances are relevant within the context of Kelly.⁹

A very risk-tolerant person might bet 100% Kelly while a risk-averse person might bet 20% Kelly.

And here’s the thing, even 100% Kelly is dumb in the real world and no real risk-taker I have ever met (e.g. Poker players, investors) ever advocates betting 100% Kelly.

Kelly represents the limit for a rational bet. Betting even one penny more than Kelly would bring increased risk, increased variance and decreased long-term profit.¹⁰

In the real world, you don’t ever know the actual odds. If you bet Full Kelly and you are off only very slightly in your estimation of the odds, you are hurting your long-term growth rate!

As the bet size approaches the Kelly-optimal point, the ratio of additional risk to additional profit goes to infinity. Eventually, you would have to risk an additional one billion dollars to earn one more cent of expected profit.

Betting a fraction of Kelly decreases your variance at a greater rate than it decreases your profit. For instance:

• Betting 50% of Kelly returns 75% of the Kelly-optimal profit with only 1/4th of the variance.
• Betting 30% of Kelly returns 51% of the Kelly-optimal profit with only 1/11th of the variance.

When you add in psychological risk tolerance plus having some humility about the uncertainty of knowing correct probabilities in a real world scenario, fractional Kelly bet sizing makes a ton of sense.

Source: Nick Yoder on the Kelly Criterion

Half Kelly is the most aggressive number I have ever heard of a real risk taker using and I would estimate that something like 10% Kelly to 30% Kelly is what most people use in practice.

Using SBF’s example of a bet that pays 10,000 to one, a fractional kelly approach would imply betting something in the range of 1-3% of your bankroll rather than his proposed 50%.

So, if you have a high-risk tolerance, maybe you go 30% Kelly. If you have a low-risk tolerance, go 10% Kelly (or less!).

SBF’s idea of betting $50k in this scenario was well into “suicidal” territory and, well, it seems that Kelly won this round (again).

Conclusion

The assumption in utility theory that SBF presumably disagreed with was that his utility function was closer to linear and therefore he was willing to pay much more to play the game.

Many investors make a similar mistake.

I’ll pick on the crypto bros because the same math is a pretty good example. If I were to tell many crypto investors that some crypto token had a 10,000 to 1 payoff with 10% probability and ask them how much they want to wager, I can almost guarantee that many of them would say a lot more than our 1-3% allocation range. However, Kelly would suggest that is the correct allocation in optimizing your long-term wealth. In my opinion, most investors would be better served by significantly reducing their bet size on individual assets and increasing their diversification of their portfolio.

Last Updated on November 18, 2022 by Taylor Pearson

1. For a more rigorous and academic treatment of, I recommend Ole Peters paper The time resolution of the St. Petersburg paradox.
2. What was your number? The first time I heard, I thought $5 sounded too cheap and $10 too expensive so I ended up in the $6-8 range.

3. Some circumstances can render this assumption invalid (Bernoulli cites an imprisoned rich man who only needs another 2,000 ducats to buy his freedom and so is willing to take bigger risks as the difference between 1999 ducats and 2,000 is enormous for him).

4. The Copenhagen Experiment supports the conclusion that people recognize this even if they are not consciously aware of it.
5. Since most people’s eyes will glaze over if I put equations in here and I’m not any good at math to begin with, I’m just going refer you to Peters’ paper
6. Anyone that is smarter than me on this feel free to explain to me why this is wrong, but this is my impression. Also, it is worth noting that he does say “let’s assume you only get to do this bet once” which in effect does ignore the role of time and kind of makes his argument work.

   I guess you can do that, but it seems like a weird thing to do, especially for someone that is a donor the Long-Term Future Fund which “aims to positively influence the long-term trajectory of civilization by making grants that address global catastrophic risks, especially potential risks from advanced artificial intelligence and pandemics.” Long-term thinking would necessarily seem to imply a healthy respect for the roll of time.

7. You may at some point around here think something like “No shit I don’t live in multiple parallel universes, why then do you keep saying that like it’s a profound observation.” To which I would reply, “fair enough, but, alas, this is literally what is taught to millions of kids every year.”
8. Credit for this analogy to David Dredge’s excellent blog.
9. There is an interesting question about what role utility does play in the context of Kelly and ergodicity economics. There are edge cases in which it would be a bad assumption to assume someone is optimizing for long-term growth.

   Like a dramatic example would be your child was kidnapped and the ransom was 50% more than your net worth. It would be “rational” to play a game in that case which did not maximize long-run wealth because there is a weird utility curve where getting your kid back is worth infinitely more (presumably) than not. A less extreme example I have actually heard someone use is: They either wanted to make enough money to retire while their kids still lived at home or they were happy to work much longer if the risk didn’t pan out.

   What’s weird about this in the context of SBF and FTX is that he was apparently a big donor the Long-Term Future Fund which “aims to positively influence the long-term trajectory of civilization by making grants that address global catastrophic risks, especially potential risks from advanced artificial intelligence and pandemics.” You would think someone involved in that would very much thing in terms of long-term wealth growth, but apparently not?

10. See Nick Yoder’s excellent explanation of the Kelly Criterion