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When systematic advantages are gambled away.

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In one interesting experiment, participants (in the financial markets) were given a coin and funding to join the game. They were allowed to take any winnings home with them. Nevertheless, one in four managed to gamble everything away.

The Experiment

The coin toss experiment is described by Victor Haghani and Richard Dewey in their paper "Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin". [1] Haghani is well known in the financial industry: He was one of the founding partners of the now-failed hedge fund LTCM.

Before the start of the game, the 61 participants were given a clear indication that the simulated coin had a 60 percent probability of heads and a 40 percent probability of tails. The starting capital was 25 US dollars each. It was explained that the balance would be paid out at the end of the game subject to a maximum amount. The maximum was 250 US dollars and was only communicated to the players when they reached this amount in order not to distort the previous expectation.

The results

The participants in the experiment were mostly economics students and young professionals from financial companies. One could therefore assume that they should be well prepared to play a simple game with fixed positive expectations. But the results had a surprise in store for them.

coin flip experiment
Figure 1) Result of the coin toss experiment
The graph shows the distribution of the final assets after 30 minutes of playing time. Roughly speaking, three categories can be distinguished: Gamblers (left in red), conservatives (centre in yellow) and strategists with optimal betting behaviour (right in green).
Source: Haghani, V. / Dewey, R. (2016), Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin, p. 4

The participants placed suboptimal bets in all shapes and sizes: too high or too low, and showed erratic betting behaviour irrationaly betting on tails instead of heads. In this way, most of them gambled away their chance to go home with 250 dollars after half an hour of play. In fact, only 21 percent of the participants reached this maximum. But even more astonishing was the fact that 28 percent of the participants went almost or completely bankrupt (residual value below 2 dollars).

„If you gave an investor the next day′s news 24 hours in advance, he would go bust in less than a year.“ (Nassim Taleb)

Even from a purely intuitive perspective there seems to be an optimal betting strategy in this game. If the risk is too high, you run the risk of losing so much money on a series of number throws that you can't recover from it. On the other hand, if the percentage at risk is too low, you do not make the most of the temporary opportunity to place bets with a statistical advantage.

If you gave an investor the next day′s news 24 hours in advance, he would go bust in less than a year.
Nassim Taleb

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.


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?

[1] Haghani, V. / Dewey, R. (2016), Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin, Elm Partners & Royal Bridge Capital

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