Bet Sizing Is Not Intuitive

Humans are not good bettors.

It takes effort both in study and practice to become more proficient. But like anything hard, most people won’t persevere. Devoting some cycles to improve will arm you with a rare arrow in your quiver as you go through life.

Skilled betting demands 2 pivotal actions:

1. Identifying attractive propositions
   This can be coded as “positive expected value” or “good risk/reward”. There is no strategy that turns a bad proposition into an attractive one on its own merit (as opposed to something like buying insurance which is a bad deal in isolation but can make sense holistically). For example, there is no roulette betting strategy that magically turns its negative EV trials into a positive EV session.
   
2. Effective bet sizing
   Once you are faced with an attractive proposition, how much do you bet? While this is also a big topic we can make a simple assertion — bad bet sizing is enough to ruin a great proposition. This is a deeper point than it appears. By sizing a bet poorly, you can fumble away a certain win. You cannot afford to get bet sizing dramatically wrong.

Of these 2 points, the second one is less appreciated. Bet sizing is not very intuitive.

To show that, we will examine a surprising study.

The Haghani-Dewey Biased Coin Study

In October 2016, Richard Dewey and Victor Haghani (of LTCM infamy) published a study titled:

Observed Betting Patterns on a Biased Coin (Editorial from the Journal of Portfolio Management)

The study is a dazzling illustration of how poor our intuition is for proper bet sizing. The link goes into depth about the study. I will provide a condensed version by weaving my own thoughts with excerpts from the editorial.

The setup

• 61 individuals start with $25 each. They can play a computer game where they can bet any proportion of their bankroll on a coin. They can choose heads or tails. They are told the coin has a 60% chance of landing heads. The bet pays even money (i.e. if you bet $1, you either win or lose $1). They get 30 minutes to play.
• The sample was largely composed of college-age students in economics and finance and young professionals at financial firms. We had 14 analyst and associate-level employees of two leading asset management firms.

Your opportunity to play

Before continuing with a description of what an optimal strategy might look like, we ask you to take a few moments to consider what you would do if given the opportunity to play this game. Once you read on, you’ll be afflicted with the curse of knowledge, making it difficult for you to appreciate the perspective of our subjects encountering this game for the first time.

If you want to be more hands-on, play the game here.

Devising A Strategy

1. The first thing to notice is betting on heads is positive expected value (EV). If X is your wager:
   
   EV = 60% (x) – 40% (x) = 20% (x)
   
   You expect to earn 20% per coin flip. 
   
2. The next observation is the betting strategy that maximizes your total expected value is to bet 100% of your bankroll on every flip. 
   
3. But then you should notice that this also maximizes your chance of going broke. On any single flip, you have a 40% of losing your stake and being unable to continue this favorable game. 
   
4. What if you bet 50% of your bankroll on every flip?
   
   On average you will lose 97% of your wealth (as opposed to nearly 100% chance if you had bet your full bankroll). 97% sounds like a lot! How does that work?
   
   If you bet 50% of your bankroll on 100 flips you expect 60 heads and 40 tails. 
   
   If you make 50% on 60 flips, and lose 50% on 40 flips your expected p/l:
   1.5⁶⁰ x .50⁴⁰ = .033
   You will be left with 3% of your starting cash! This is because heads followed by tails, or vice versa, results in a 25% loss of your bankroll (1.5 * 0.5 = 0.75).
   This is a significant insight on its own. Cutting your bet size dramatically from 100% per toss to 50% per toss left you in a similar position — losing all or nearly all your money.

Optimal Strategy

There’s no need for build-up. There’s a decent chance any reader of this blog has heard of the Kelly Criterion which uses the probabilities and payoffs of various outcomes to compute an “optimal” bet size. In this case, the computation is straightforward — the optimal bet size as a fraction of the bankroll is 20%, matching the edge you get on the bet.

Since the payoff is even money the Kelly formula reduces to 2p -1 where p = probability of winning.

2 x 60% – 1 = 20%

The clever formula developed by Bell Labs researcher John Kelly:

provides an optimal betting strategy for maximizing the rate of growth of wealth in games with favorable odds, a tool that would appear a good fit for this problem. Dr. Kelly’s paper built upon work first done by Daniel Bernoulli, who resolved the St. Petersburg Paradox— a lottery with an infinite expected payout—by introducing a utility function that the lottery player seeks to maximize. Bernoulli’s work catalyzed the development of utility theory and laid the groundwork for many aspects of modern finance and behavioral economics. 

The emphasis refers to the assumption that a gambler has a log utility of wealth function. In English, this means the more money you have the less a marginal dollar is worth to you. Mathematically it also means that the magnitude of pain from losing $1 is greater than magnitude of joy from gaining $1. This matches empirical findings for most people. They are “loss-averse”.

How did the subjects fare in this game?

The paper is blunt:

Our subjects did not do very well. Suboptimal betting came in all shapes and sizes: overbetting, underbetting, erratic betting, and betting on tails were just some of the ways a majority of players squandered their chance to take home $250 for 30 minutes play.

Let’s take a look, shall we?

Bad results and strange behavior

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%

• 1/3 of the participants finished will less money than the $25 they started with. (28% went bust entirely!)
• 67% of the participants bet on tails at some point. The authors forgive this somewhat conceding that players might be curious if the tails really are worse, but 48% bet on tails more than 5 times! Many of these bets on tails occurred after streaks of heads suggesting a vulnerability to gambler’s fallacy.
• Betting patterns and debriefings also found prominent use of martingale strategies (doubling down after a loss).
• 30% of participants bet their entire bankroll on one flip, raising their risk of ruin from nearly 0% to 40% in a lucrative game!

Just how lucrative is this game?

Having a trading background, I have an intuitive understanding that this is a very profitable game. If you sling option contracts that can have a $2 range over the course of their life and collect a measly penny of edge, you have razor-thin margins. The business requires trading hundreds of thousands of contracts a week to let the law of averages assure you of profits.

A game with a 20% edge is an astounding proposition.

Not only did most of our subjects play poorly, they also failed to appreciate the value of the opportunity to play the game. If we had offered the game with no cap [and] assume that a player with agile fingers can put down a bet every 6 seconds, 300 bets would be allowed in the 30 minutes of play. The expected gain of each flip, betting the Kelly fraction, is 4% [Kris clarification: 20% of bankroll times 20% edge].

The expected value of 300 flips is $25 * (1 + 0.04)³⁰⁰ = $3,220,637!

In fact, they ran simulations for constant bet fractions of 10%, 15%, and 20% (half Kelly, 3/4 Kelly, full Kelly) and found a 95% probability that the subjects would reach the $250 cap!

Instead, just over 20% of the subjects reached the max payout.

Editorialized Observations

• Considering how lucrative this game was, the performance of the participants is damning. That nearly one-third risked the entire bankroll is anathema to traders who understand that the #1 rule of trading (assuming you have a positive expectancy business) is survival.
  
• Only 5 out of the 61 finance-educated participants were familiar with Kelly betting. And 2 out of the 5 didn’t consider using it. A game like this is the context it’s tailor-made for!
  
• The authors note that the syllabi of MIT, Columbia, Chicago, Stanford, and Wharton MBA programs do not make any reference to betting or Kelly topics in their intro finance, trading, or asset-pricing courses. 
  
• Post-experiment interviews revealed that betting “a constant proportion of wealth” seemed to be a surprisingly unintuitive strategy to participants. 
  

Given that many of our subjects received formal training in finance, we were surprised that the Kelly criterion was virtually unknown among our subjects, nor were they able to bring other tools (e.g., utility theory) to the problem that would also have led them to a heuristic of constant-proportion betting. 

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? Given the propensity of our subjects to bet on tails (with 48% betting on tails on more than five flips), is it any surprise that people will pay for patently useless advice? What do the results suggest about the prospects for reducing wealth inequality or ensuring the stability of our financial system? Our research suggests that there is a significant 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.

Our research will be worth many multiples of the $5,574 winnings we paid out to our 61 subjects if it helps encourage educators to fill this void, either through direct instruction or through trial-and-error exercises like our game. As Ed Thorp remarked to us upon reviewing this experiment, “It ought to become part of the basic education of anyone interested in finance or gambling.”

Extra thoughts

I will add my own concern. It’s not just individual investors we should worry about. Their agents in the form of financial advisors or fund managers, even if they can identify attractive propositions, may undo their efforts by poorly sizing opportunities by either:

1.  falling far short of maximizing
   
   Since great opportunities are rare, failing to optimize can be more harmful than our intuition suggests…making $50k in a game you should make $3mm is one of the worst financial errors one could make.
   
2. overbetting an edge
   
   There isn’t a price I’d play $100mm Russian Roulette for

Getting these things correct requires proper training. In Can Your Manager Solve Betting Games With Known Solutions, I wonder if the average professional manager can solve problems with straightforward solutions. Nevermind the complexity of assessing risk/reward and proper sizing in investing, a domain that epitomizes chaotic, adversarial dynamics.

Nassim Taleb was at least partly referring to the importance of investment sizing when he remarked, “If you gave an investor the next day’s news 24 hours in advance, he would go bust in less than a year.”

Furthermore, effective sizing is not just about analytics but discipline. It takes a team culture of truth-seeking and emotional checks to override the biases that we know about. Just knowing about them isn’t enough. The discouraged authors found:

…that without a Kelly-like framework to rely upon, our subjects exhibited a menu of widely documented behavioral biases such as illusion of control, anchoring, overbetting, sunk-cost bias, and gambler’s fallacy.

Conclusion

Take bet sizing seriously. A bad sizing strategy squanders opportunity. With a little effort, you can get better at maximizing the opportunities you find, rather than needing to keep finding new ones that you risk fumbling.

You need to identify good props and size them well. Both abilities are imperative. It seems most people don’t realize just how critical sizing is.

Now you do.