A recent betting experiment among finance students and professionals based on biased coin flipping revealed a wide gap between rational and actual behavior. The optimal strategy, which would have been constant and moderate risk taking (“Kelly criterion”), was not widely applied, notwithstanding education and training in finance. Instead, the experiment revealed a range of common behavioral biases. It challenges the general assumption of rational decision-making of finance professionals under uncertainty.

Haghani, Victor and Richard Dewey (2016), “Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin”, Working Draft, October 19, 2016

The post ties in with our lecture on information efficiency, in particular the lack of plausibility of fully efficient markets.

The below are excerpts from the paper. Headings and some other cursive text has been added for context and convenience of reading.

The experiment

“Our coin-flipping experiment was played by 61 subjects, in groups of 2-15…The sample was largely comprised of college age students in economics and finance and young professionals at finance firms… We programmed the coin to be in a  flipping mode for about 4 seconds, to create some suspense on each flip, and also to limit the number of flips to about 300.”

“Prior to starting the game, participants read a detailed description of the game, which included a clear statement, in bold, indicating that the simulated coin had a 60% chance of coming up heads and a 40% chance of coming up tails. Participants were given $25 of starting capital and it was explained in text and verbally that they would be paid, by check, the amount of their ending balance subject to a maximum payout. The maximum payout would be revealed if and when subjects placed a bet that if successful would make their balance greater than or equal to the cap. We set the cap at $250, ten times the initial stake.”

The Kelly criterion

“The Kelly criterion [is] a formula [that] provides an optimal betting strategy for maximizing the rate of growth of wealth in games with favorable odds…Only 5 of our 61 financially sophisticated students and young investment professionals reported that they had ever heard of the Kelly criterion.”

“The basic idea of the Kelly formula is that a player who wants to maximize the rate of growth of his wealth should bet a constant fraction of his wealth on each flip of the coin, defined by the function 2 * p – 1, where p is the probability of winning. The formula implicitly assumes the gambler has log utility. It is intuitive that there should be an optimal fraction to bet; if the player bets a very high fraction, he risks losing so much money on a bad run that he would not be able to recover, and if he bet too little, he would not be making the most of what is a finite opportunity to place bets at favorable odds. While it is true that the expected value of the game goes up the higher the fraction the player bets, the outcomes become so skewed that a player who exhibits risk aversion will find an optimal betting fraction well below 100%. The odds themselves play a role in the optimal fraction to bet; the more favorable the odds, the higher a fraction one ought to bet. Finally, as the flips are independent random outcomes, the strategy should only depend on the player’s account balance, and not on the pattern of previous flips.”

“In our game, the Kelly criterion would tell the subject to bet 20% (2 * 0.6 – 1) of his account on heads on each flip. So, the first bet would be $5 (20% of $25) on heads, and if he won, then he’d bet $6 on heads (20% of $30), but if he lost, he’d bet $4 on heads (20% of $20), and so on… The expected gain of each flip, betting the Kelly fraction, is 4% and so the expected value of 300 flips is…$3,220,637!

The results

“Our subjects did not do very well. While we expected to observe some sub-optimal play, we were surprised by the pervasiveness of it.”

Only 21% of participants reached the maximum payout of $250, well below the 95% that should have reached it given a simple constant percentage betting strategy of anywhere from 10% to 20%…One third of the participants wound up with less money in their account than they started with. More astounding still is the fact that 28% of participants went bust and received no payout…The average ending bankroll of those who did not reach the maximum and who also did not go bust, which represented 51% of the sample, was $75. While this was a tripling of their initial $25 stake, it still represents a very sub-optimal outcome.”


“Of the 61 subjects, 18 subjects bet their entire bankroll on one flip, which increased the probability of ruin from close to 0% [based on the Kelly criterion] 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.”

“The average bet size across all subjects was 15% of the bankroll, so participants bet less, on average, than the Kelly criterion fraction, which would make sense in the presence of a maximum payout that would be within reach. However, this apparent conservatism was completely undone by participants generally being very erratic with their fractional betting patterns, betting too small and then too big.”

“We observed 41 subjects (67%) betting on tails at some point during the experiment…29 players (48%) bet on tails more than 5 times in the game.”

The lessons

“The straightforward notion of taking a constant and moderate amount of risk and letting the odds work in one’s favor just doesn’t seem obvious to most people…The Kelly Criterion was virtually unknown.”

“Our subjects exhibited a menu of widely documented behavioral biases such as illusion of control, anchoring, over-betting, sunk-cost bias, and gambler’s fallacy.”

Brief definitions:

  • Illusion of control is the tendency for human beings to believe they can control or at least influence outcomes that they demonstrably have no influence over. Academic research has shown that people often behave as if chance events are accessible to personal control.
  • Anchoring describes the common human tendency to rely too heavily (or anchor) on one piece of information when making decisions. Individuals fixate on specific information and then adjust to that information to account for other elements of the circumstance.
  • The sunk-cost bias is the tendency of people to carry on with a strategy that is not meeting expectations because of the time or money they have already spent on it. It is a form of evidently irrational decision making, because it is based on irrelevant data.
  • Gambler’s fallacy is the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future. More generally, it is the logical fallacy that a random process becomes less random, and more predictable, as it is repeated.

“There is a meaningful gap in the education of young finance and economics students when it comes to the practical application of the concepts of utility and risk taking. The existence of this gap is even more surprising than the poor play of our subjects. After all, can we really blame them if they haven’t received sufficient practical training?”