When systematic advantages are gambled away.

The Kelly formula

The solution to the betting problem is the classic Kelly formula, 2 * p - 1, where p represents the probability of winning. It is much simpler than just about any other formula in the financial world, but still unknown to most market participants. Only five of the 61 participants said they had heard of it before. This is despite the fact that John Kelly developed the formula in 1955 and that it is obviously relevant in the financial sector.

The formula calculates the optimum constant proportion of available capital to be used to maximise asset growth in games with statistically advantageous odds. For the example in the paper this means: Ideally, participants should bet 2 * 0.6 - 1 = 20 per cent of the available capital on heads for each coin toss. For the first toss, this would be 5 dollars (20 percent of 25 dollars). If it is a win, the next roll should be 6 dollars (20 percent of 30 dollars). If it is a loss, on the other hand, it would be 4 dollars (20 percent of 20 dollars). And so on.

A player who does not need to think and is fast could make a coin toss every six seconds, i.e. make a total of 300 passes in the 30 minutes (on average, the participants only made 120). The authors calculate that the expected win of each toss is four percent if the optimal Kelly proportion is set, exemplary for the first toss: 60 percent * 5 dollars - 40 percent * 5 dollars = 1 dollar = 4 percent of 25 dollars. With 300 throws this would correspond to an expected value of 25 * 1.04^300 = 3,220,637 dollars. This also makes it clear why the authors had to provide for a maximum amount in their experiment to be on the safe side!

Interviews following the experiment showed that some participants considered a doubling or martingale strategy, in which the amount of the stakes is increased after losses, to be optimal instead of the constant percentage stake. Another popular approach was small and constant stakes, obviously to reduce the risk of ruin and maximise the probability of winning - but at the considerable expense of the size of the winnings.

Conclusion

The authors of the paper write that they had expected some ill-conceived betting strategies from the participants. However, it was not foreseen that 28 percent would almost or completely go bust, even though they had a clear statistical advantage - especially considering that most of the participants had formal training in the financial sector.

It is sobering that only a few participants had heard of the Kelly formula before, but also that most participants apparently did not have the analytical know-how to come to the conclusion themselves that a constant percentage betting stake should be optimal. Without such a rational framework, the door was open to all kinds of behavioural effects observed in the experiment: Control illusion, anchoring, excessive stakes, sunk cost bias and Gambler's Fallacy.

And there is another question: If a high proportion of quantitatively trained and financially literate people have such great difficulty in winning a simple game with a winning coin, what does this mean for the prospects of the broad mass of the population to invest their savings wisely in the long term in the far more complex reality?