Lessons from Betting on a Biased Coin: Cool heads and cautionary tales


What would you do if you were invited to play a game where you were given $25 and allowed to place bets for 30 minutes on a coin that you were told was biased to come up heads 60% of the time? Well, we programmed up a biased coin and did just that, inviting 61 young, quantitatively trained men and women to play this game, and we paid them whatever they had left after 30 minutes of betting, which amounted to over $5,000 in aggregate.

Here are some of the main takeaways from the research paper we wrote on this experiment, which was just accepted in SSRN’s elibrary (click here for the full 7 page paper):

In a nutshell, the majority of these 61 players managed their betting very sub-optimally and squandered an opportunity to take home $250, which they were almost certain to win if they followed one of a range of simple and disciplined (and boring) strategies that allowed the odds to work in their favor. Evidently, these strategies weren’t as obvious or intuitive to our subjects as we thought they would be. Their betting displayed as many behavioural and cognitive biases as you can shake a stick at.

‘How did you go bankrupt? Gradually, and then suddenly.’ (Ernest Hemingway, The Sun Also Rises 1926)

Yes, we expected we’d see some suboptimal betting, but the depth and breadth of poor play was impressive. About 30% of the subjects actually went bust, losing their full $25 stake.

The note also discusses optimal betting strategies (in particular the Kelly criterion), the amount someone should be willing to pay to play the game (much less than the game’s expected value) and the ways in which the game is similar to investing in the stock market. You’ll see why we think that a modified version of our game would be an effective teaching aid. It certainly would be better than investment tournaments, where participants are taught all the wrong lessons about investing- after all, no one ever won an investment competition by putting 70% of their play capital in Vanguard’s total world market index fund.

The results of our study raise important questions. If a high fraction of quantitatively sophisticated, financially trained individuals have so much difficulty in playing a simple game with a biased coin, what should we expect when it comes to the more complex and long-term task of investing one’s savings? Is it any surprise that people will pay for patently useless advice? [1] What do the results of this experiment say about the prospects for reducing wealth inequality, or ensuring the stability of our financial system?

In the words of Ed Thorp, who reviewed and helped guide our research:

This is a great experiment for many reasons.  It ought to become part of the basic education of anyone interested in finance or gambling.

We hope you’ll read the paper, try out the game (for educational purposes, no cash payouts), and, if you find it interesting, pass it along to your family and friends.

UPDATE: You can also now read about the research paper in The Financial Times, The Economist and Bloomberg.

[1] As documented in studies like Powdthavee, Nattavudh and Riyanto, Yohanes, E., 2012 Why Do People Pay for Use-less Advice? Implications of Gamblers and Hot-Hand Fallacies in False-Expert Setting. Working Paper (IZA DP No. 6557)