Biased Coin Study

A Xmas Day hit count of 915 might not have been quite the best present ever, (that would have been my claret and blue Crystal Palace socks in 1969), but it was a surprise that so many people stopped by after attending church services and opening Santa's presents.

I know the Queen came up with a good excuse to miss church, and there were a number of hits from the Sandringham area on Xmas morning, but I'm not one to speculate.

Courtesy of the Sports Trader blog, I did read an interesting post on a 'biased coin' study, (Victor Haghani , Richard Dewey, 2016) which found that even when told a coin would come up heads 60% of the time, a group of 61 people managed to significantly under-perform expectations over a 30 minute period. 

The sample was largely comprised of college age students in economics and finance and young professionals at finance firms. We had 14 analyst and associate level employees at two leading asset management firms. The sample consisted of 49 males and 12 females. Our prior was that these participants should have been well prepared to play a simple game with a defined positive expected value.

There were a number of surprises in the study, not least that of the 61 financially sophisticated students and young investment professionals involved, only five had ever heard of the Kelly criterion, and:

Interestingly, having heard of Kelly did not seem to help two of them: one barely managed to double his stake, and the other one only broke even after about 100 flips.

28 people managed to lose their starting stake of $25, and only 21 managed to win the maximum sum (capped at $250), significantly fewer than the the "95% that should have reached it given a simple constant percentage betting strategy of anywhere from 10% to 20%."

Some other highlights for me were that

of the 61 subjects, 18 subjects bet their entire bankroll on one flip, which increased the probability of ruin from close to 0% using Kelly to 40% if their all-in flip was on heads, or 60% if they bet it all on tails, which amazingly some of them did.

And then there was this:

Betting patterns and post experiment interviews revealed that quite a few participants felt that some sort of doubling down, or Martingale betting strategy, was optimal, wherein the gambler increases the size of his wagers after losses.

We observed 41 subjects (67%) betting on tails at some point during the experiment. Betting on tails once or twice could potentially be attributed to curiosity about the game, but 29 players (48%) bet on tails more than 5 times in the game. It is possible that some of these subjects questioned whether the coin truly had a 60% bias towards heads, but that hypothesis is not supported by the fact that within the subset of 13 subjects who bet on tails more than 25% of the time, we found they were more likely to make that bet right after the arrival of a string of heads. This leads us to believe that some combination of the illusion of control, law of smallnumbers bias, gamblers fallacy or hot hand fallacy was at work. 

Even though you won't win any real money, you can try the experiment here if you have 30 minutes of your day free, but even going in armed with a full understanding of Kelly isn't a guarantee of success! I found myself questioning whether Heads really did have a 60% probability after sequences such as those pictured, though I have no reason to doubt the claim. 

In the conclusion, and the study is well worth reading in full, the authors write:

These results 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, as documented in studies like Powdthavee (2012)? What do the results of this experiment say about the prospects for reducing wealth inequality, or ensuring the stability of our financial system? 

As mentioned here many times, don't pay for financial advice - use low cost index funds and time to make your fortune! 

When it comes to sports betting, where our edge is typically small at best, and likely to be of limited duration, a losing run is psychologically even more devastating. 

One edge that shows signs of finally dissipating this season is that of the Bundeslayga System. In Bundeslayga.1 results are still positive (+6.89%), but Bundeslayga.2 is headed for a first losing season with an ROI of -9.18%.