ARE WE ANY GOOD at correctly analyzing simple financial situations involving probabilities? Kenyon, my brother and fellow HumbleDollar contributor, introduced me to a 2016 study that suggests that many of us are shockingly poor at doing so.
Sixty-one business students and young professionals at financial firms were presented with the following scenario: At a website, you’ll be given $25 and allowed to bet on a computer-generated coin flip. You may bet on either heads or tails. It isn’t a fair coin. With each flip, there’s a 60% chance of heads and a 40% chance of tails. If you win the bet, the amount you wagered will be added to your kitty. If you lose, it will be subtracted. With each bet, you may wager any sum up to the amount you have. You have 30 minutes. Your goal is to end with the largest amount possible, which you’ll then receive, subject to a $250 maximum.
What strategy would you follow? How would you fare?
If you want to find out how, don’t read beyond this paragraph until you’ve first tried an abbreviated version of this experiment. Instead of 30 minutes, you’ll be given 10 minutes. Also, you won’t receive your ending balance. Otherwise, the situation is as described above. Click on this link and try it. In the comments section below, feel free to post the strategy you followed and your ending balance.
Here are three possible strategies:
The investigators stated that “[w]hile we expected to observe some sub-optimal play, we were surprised by the pervasiveness of it.” That’s an understatement.
A person should never bet on tails, and yet 67% of the participants bet on tails at least once. Nearly half the players (48%) bet on tails more than five times. One player in five (21%) bet on tails at least a quarter of the time.
It might be rational to bet on tails if you believed the experimenters lied when they said the computer had been programmed so there’s a 60% chance of coming up heads. The only other possible reason for betting on tails: You think past performance had some value in predicting future performance. After a string of heads, some people might believe tails is bound to be next. But in this experiment, each flip had a 60% chance of coming up heads. No one should ever bet on tails.
Surprisingly, 28% of participants went bust, which the experimenters defined as ending with less than $2. A person should never bet so much that they have a 40% chance of ending up with nearly nothing.
The authors assumed that 95% of the participants would reach the $250 maximum. In reality, only 21% reached this goal. As shown below, following an optimal strategy, you should have about $8,973 after 30 minutes. Thus, even with a lot of sub-optimal bets, a person should still reach $250 after 30 minutes. Yet four out of five participants failed to achieve this.
Based on some reasonable assumptions, the researchers suggest a bet of 20% of the current balance is the optimum bet. Why? Those who are math-phobic can skip the equations below.
But for my fellow nerds, here’s the mathematical explanation: With a 20% bet, the expected value of each flip is a 4% increase in your kitty. Why? You have a 60% chance of a 20% gain, and a 40% chance of a 20% loss, which mathematically looks like this:
(0.6 x 0.2) – (0.4 x 0.2) = 0.12 – 0.08 = 0.04
The outcome is highly dependent on the number of flips. If there are 150 flips during the 30 minutes, a player should have $8,973:
$25 x (1.04)150 = $25 x 358.92 = $8,973
What should we expect during 10-minute experiments? If people follow the strategy of betting 20% of the balance and they make 50 coin flips, they should have $178:
$25 x (1.04)50 = $25 x 7.11 = $178
I’m not including the math, but the $250 maximum for the 30-minute experiment scales down to $54 for the 10-minute version. While the optimal betting strategy yields $178 in 10 minutes, anyone who reaches at least $54 has been somewhat successful.
My brother and I independently tried the 10-minute coin flip experiment. We had absolutely no prior discussion about strategies. He and I both bet a constant percentage of the balance. He decided to use 25% and I used 20%. In 10 minutes, he had $68 and I had $134, both very respectable.
I shared this game with three college professor colleagues—two business professors and one psychology professor who teaches statistics. Two of the three went bust. Those two both used a constant percentage strategy, but they chose percentages that were too high. A string of tails wiped them out. The third ended with $560. Using a modified 20% strategy, he had $280 with just seconds to go. He bet it all and won.
If most well-educated people have trouble with this straight-forward proposition, it’s hardly surprising that the general population has trouble navigating the myriad choices—with so many unknowns involved—when saving and investing for retirement.
Larry Sayler is the only person with a Wharton MBA who also graduated from Ringling Bros. and Barnum & Bailey’s Clown College. Earlier in his career, he served as CFO for three manufacturing and service organizations. For 16 years before his retirement, Larry taught accounting at a small Christian college in the Midwest. His brother Kenyon also writes for HumbleDollar. Check out Larry’s earlier articles.