Larry Sayler

**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:

- Bet a constant percentage of your balance. But how much? If the percentage is too low, you won’t win very much. If it’s too high, you risk getting wiped out with a few losing bets.
- Bet a constant dollar amount. As above, if the amount is too small, your bets won’t add much when you win. But if it’s too large, you run the risk of losing everything with a few consecutive losses.
- Double your bet after any loss, also known as doubling down. This method guarantees a profit when you eventually win, but there are two important qualifications. First, it assumes you don’t run out of money. In this game, you may quickly run out of money. Second, it assumes there’s no dollar limit on how much you bet each time.

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.*

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DI must be missing something here, but how is it possible to get wiped out with a constant percentage? If you’re not betting 100%, then, by definition, you can never get to $0 (unless you can’t subdivide $0.01 maybe?).

You’re right, you can’t get wiped out by using a constant percentage strategy. But here’s what Larry wrote about the experiment: “Surprisingly, 28% of participants went bust, which the experimenters defined as ending with less than $2.”

Great contribution. ( I still don’t understand how so many of these select people did so poorly….)

A very well shown, and analyzed contribution.

Seems again overall sentiment supports Taylor Larimore and the Bogleheads adage.

‘Its not timing the market, but time in the market’

that usually pays unexpectedly well over time.

So how does this apply to allocating a portfolio of stocks vs fixed income? With a binary 60/40 rule, we know the odds. But stocks could go up or down at any moment and the new built in breakers won’t help those of us in mutual funds who don’t trade all the time.

I am going to model this later with an iterative computer code. The math you present looks sound to find optimal percentage except the final fixed amount doesn’t appear to include compounding.

I am surprised that you did not mention the Kelly Criterion. From Wikipedia: “the Kelly criterion leads to higher wealth than any other strategy in the long run.”

Flipped more than 90 times, always betting $3 on heads. My balance was down to $13 four times, up to $43 twice, and ended at $28.

I know that with probabilities, the number of attempts matter, with more being better. But it was hard to keep going when I lost several flips in a row and saw my balance dropping, dropping, dropping. I suspect younger investors have an even harder time sticking with plan when portfolio values are declining.

Thanks for the post and the experiment. I may try it again.

I won $500 with about 15 tosses, using both heads and tails, mostly heads of course.

Jack, great question. I will review the math for this and post a response. Unfortunately, it will probably be tomorrrow, not today

OK, thanks!

I enjoyed this article. How would you calculate the optimal percentage to bet in order to maximize gains?

Jack, I was wondering the same thing. I’m not sure how to do a close-form solution, but I was thinking a spreadsheet simulation would not be too hard to build.

Thanks Rick.

Larry, thanks for a fun article. It hits on some key notions of probability, namely independence and expected return. A decent understanding of these is very useful.

I started with $5 bets (20%), and stuck with tails. my biggest challenge was sticking with he exercise for 10 minutes, and not making typo/mousing mistakes! This demonstrates why I’m a buy and hold indexer. I’m not very interested in frequent trading, and stress out too much when i win. Others i know love it and have done well, but I wasn’t made for it. Probably why I never go to a casino or bet online.

I got to $325 by betting roughly 10% each round, always heads. With the odds in your favor, probably even more important than betting the optimal percentage is to play as many rounds as possible. Therefore I tried to make the clicks fast and even round the amounts to bet faster. I got to play 120 rounds vs an average of about 50 according to the chart elm shows as feedback.

Smart move!

Interesting! I actually stopped playing after tripling my starting amount, about 4 minutes into the session. Even knowing it was stacked in my favor, it was difficult to risk the gains. Note: I put my mindset into the game’s top limit of $250 limit. No need to swing for the fences. It felt good to quit while ahead!

Sounds exactly like my rebalancing practices 😀!

Thanks for a very fun way to expose poor thinking, Larry, and thanks to John for a great application of the lesson.

The stock market increases in ~70% of years (flips) and decreases in ~30% of flips. With a constant percentage 60% bet of the balance (60-40 or 70-30 portfolio), the expected value of each flip is a several % increase in your kitty. The outcome is highly dependent on the number of flips (years), but with enough flips (years of compounding), all coin flippers should be well ahead. Since the stock market is more skewed to positive years than the coin flips, the percentage bet should clearly be higher.

Depends on which time unit (days, months, years, decades, etc.) you are looking at for each “flip” of course.

Also, markets are less of a coin flip in that 100% “betted” (invested) will not double or go to zero on each “flip”.