The book Going Infinite by Michael Lewis is fast emerging as the go-to psychological portrait of Sam Bankman-Fried, the founder of crypto-currency company FTX who was convicted last November of fraud and other crimes. Lewis traces Bankman-Fried’s reckless betting style, which eventually led to the implosion of FTX, at least in part to a spell early in his career as a trader. Bankman-Fried’s principal learning appears to have been that one should always take a bet that has a positive expected value (i.e., if the probability of winning times the amount to be won is greater than the sum wagered, that bet has positive expected value, and should be played). He also learned to take pride in the fact that he was more willing to do this than most other people.
Quite how willing he was to take risky bets became very public during his recent court testimony, where it emerged that Bankman-Fried once told co-workers that he would always take a bet with a 50% chance of destroying the world if it offered a 50% chance of making it twice as good. This statement may look insane (it is), but it is only an extreme example of a miscalculation that we humans commonly make: we under-account for the probability of going bust.
Some 20 years after Long-Term Capital Management collapsed, one of its founders, Victor Haghani, designed a game to study how finance people manage risk. We credit Marc Rubinstein, author of the excellent newsletter Net Interest, for bringing this experiment to our attention. The game – for which Haghani recruited 61 economics students and young finance professionals – consists of a repeated coin flip over a 30-minute period. The coin is weighted so that it comes up heads 60% of the time and tails the other 40%. The rules are transparent: everybody knows that the coin is weighted, and that the time is fixed. An efficient player can rack up about 300 flips in 30 minutes, making this a clear example of what, in game theory, is called a repeated game. Beginning with a US$25 bankroll, the aim is to maximise your winnings. You can have a go yourself: versions of Haghani’s game are available online.
The game offers a clear edge to those who bet on heads. Haghani calculated that, if played optimally, the maximum payout is over US$3.2 million, though he prudently capped winnings at US$250. For a simple game of this type, where the player’s betting edge is clearly known, there is a well-known formula, known as the Kelly formula, which determines the percentage of a player’s bankroll (i.e., of the US$25) that they should bet. The formula is 2p-1, where p is the probability of winning. In this case, with a 60% probability of winning, the formula gives 2*0.6-1 = 0.2, meaning that a player should bet about 20% of their bankroll on each coin toss. It is interesting how small a proportion of the bankroll this is: anything above 20% constitutes ‘over-betting’ and is detrimental to returns. This is because the losses would be too big to subsequently be made up by the edge you have.
In repeated games, survival – with manageable losses – matters more than maximising gains in the very next coin flip. In investing, this has a useful parallel with drawdowns: the size of the drawdown you risk taking has to be proportionate to the types of returns you think you can generate. Survival enables the maximisation of long-term compounding, and counterintuitively means your bets must get smaller in absolute amounts as you lose money: 20% of US$25 is US$5, but if you lose, the next bet should be 20% of US$20, i.e., US$4. ‘Hail Marys’ rarely work.
The Kelly criterion is not something that people, even finance students, necessarily carry around in their brain. Even so, Haghani was surprised by how far from this optimal betting pattern most students landed:
- Only 21% of participants reached the maximum payout (US$250, not US$3.2 million), meaning that the widespread tendency was to over-bet. In case you are wondering, under-betting does not explain it: betting 10% of your bankroll still allows you to reach the winnings cap 95% of the time.
- 28% of players went bust. They really over-bet.
- The rest, the middle 51% that neither maxed out nor went bust, finished on average at US$75. That seems like a good return (tripling your money). But when the potential return is US$3.2 million, it definitely counts as an under-achievement.
- Probably the most interesting (to us) reason for the mediocre performance of most participants is that two-thirds of players bet on tails, with only a 40% probability of winning, at least once. 29 players (47.5%, almost half) bet on tails more than five times. This is puzzling: with such a clearly disclosed edge, there is never a good reason to bet on tails, so why do it? Boredom? The heuristic that “Tails hasn’t come up in a while”? “I’m feeling lucky”? “Tails never fails”?
Given that most participants were students or young professionals, the game may tell us something in particular about the risk-seeking behaviour of young people. But there are lessons for everyone who participates in the ‘repeated game’ of investing. To be clear, the Kelly criterion does not capture the investing environment with sufficient nuance: the concept of a disclosed and knowable edge does not exist in investments; even when trying to quantify an edge, investors reliably over-estimate it; and, of course, investing is not as binary as the double-or-bust outcome distribution of a coin flip.
Nevertheless, investors should firstly recognise that people have a clear tendency to over-bet (i.e., size their bets too large), which in combination with the fact that they overestimate their edge, is plainly dangerous. Most investment managers would kill for the guaranteed 60/40 edge of the coin-flip game, and yet, even with that perfectly transparent edge, the optimal bet size is ‘only’ 20%. Secondly, while precisely estimating an edge is impossible, there are ways to improve the probability of success when investing. Choosing companies with enduring business models, for example, or insisting on a low valuation. Also, avoiding high debt levels. Doing all three is a good idea, and if you are going to compromise, at least size those bets smaller.
Finally, we should acknowledge that the long-term compounding of repeated games does not mesh well with our intuitive capacity. Given the choice, people tend to act as if investing is a ‘one-and-done’ exercise, a tendency exacerbated by the daily availability of market prices. They see patterns where none exist, and are tempted to try to exploit them, taking bets that carry an unacceptably high probability of failure and that they would not take if they stopped to account for the fact that they will have to make up the loss in the next stages of the game. Either because they get bored or because they think they are really clever, or because their heuristics trick them, people will occasionally take short-term bets that do not make logical sense in the long run. When it comes to markets, and their perpetual tendency to behave oddly, we think this explains a lot about a lot.
