Moontower #166

One of my deepest held beliefs is that our need for coherence is a profound source of misery. We agitate for universal theories to tie everything together. Our obsessions with gurus, religion, ideology, macro, or even astrology are symptoms. We search for meaning as if it is something that’s “out there” to be discovered. I’m not holding my breath.

These quests are actively destructive when taken too seriously. When people become overly invested in any of these expeditions, we are sentenced to watching the mental gymnastics routines their precious egos cling to. It’s worse than just being cringe. It’s insidious. They dehumanize opposition so they don’t even have to consider reasonable antagonistic stances.

I just picked up Simone de Beauvoir’s book The Ethics Of Ambiguity because its description vibrates with my own feelings. I’ll report back after reading it.  (See How The Need For Coherence Drives Us Mad to see if you’d be interested in reading it.)

In the meantime, I’ll share a technique that I use to resist the seduction of coherence.

A Drawer Of Curiosities

In my notes, I keep an ever-growing list of “tensions” and “paradoxes” that I encounter from reading or experience. It is a constant reminder that every bit of advice you’ve ever heard is not universal. My buddy Jake likes to say that seat belts are the only free lunch. To which I respond, “unless the presumption of safety encourages drivers to speed or drive more recklessly”. Let’s be blunt. My response is utter ankle-biting tediousness (what I call tedious some people consider their personality). The larger takeaway is there are paradoxes running loose everywhere and if we run around trying to corral them with some ill-conceived notion that it makes us “more right” or there are truths we can somehow own and wield, then we’ve done nothing but build intellectual totems to hubris.

Instead of trying to resolve the paradoxes, maybe just accept them. Name it to tame it, put it in a drawer, and move on. You don’t need the world to bend around your own brain to protect your ego. You can just have a big list that reminds you that the task is futile. That’s the antidote.

[See A Drawer Of Curiosities for excerpts from a couple of studies that speak to the benefits of acknowledging and living with paradox. I have long thought this was important and was discussing it with a friend who said there’s actually some biology behind our resistance of paradox. They sent me the links found in the appendix of that post.]


I keep another type of list that is unexpectedly satisfying. A graveyard of ideas and projects that I’ve abandoned. It’s a form of closure. It’s a form of loving and losing. There’s no need for shame or regret because you didn’t learn to play the guitar or start that business. Most text editors have a “strikethrough”. Use it.

The things you actually did instead were a filter. They revealed your priority. (If you have a problem with your priorities that’s a separate issue). By acknowledging that you will only execute on a fraction of your ideas, you lower the stakes of having ideas, and the more ideas you allow yourself to have, the more fun life will be.

A close-minded young person feels tragic. But weirdly, I think it’s even more tragic for older people whose years give them a perch to see how wide a range of experience exists across the world and its people — and then they ignore that information. It’s like locking yourself in a room with 1 friend, setting the thermostat to your preferred temperature, throwing away the key, and talking about the same old shit until you die.

Take chances intellectually and in life. List them. If there’s nothing on the list, maybe fix that. You still have time before you yourself are in a graveyard.

Scavenger Hunt

In How to turn problems into a curiosity engineAnn-Laure Le Cunff describes a fun game renowned physicist Richard Feynman played as he navigated life:

One of Feynman’s most enduring characteristics was that he loved problems. Instead of avoiding them or trying to solve them as fast as possible, he would seek interesting problems, keep them in mind, let them simmer, and constantly try to connect his everyday experiences to these big questions.

“You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps.”

Similar to Alice who discovers a strange world through the looking glass, the questions you choose to keep in mind act as a mirror that reflects the world around you and makes you look beyond the surface of the glass.

Your favorite problems form a prism that separates incoming information into a spectrum of ideas — a frame that allows you to deliberately filter distractions, direct your attention, and nurture your curiosity. In short, your favorite problems become a curiosity engine.

Creating a list of favorite problems offers many benefits:

  • Turn stressful situations into intriguing problems to explore
  • Filter information based on whether it relates to one of your favorite problems
  • Connect with fellow curious minds who are interested in similar problems
  • Focus your attention on ideas that arouse your curiosity
  • Notice relevant patterns and potential solutions across seemingly unrelated topics

Let’s ignore yet another advice tension (Feynman is unknowingly inviting people to double-down on confirmation bias) to give this idea respect. Feynman is saying “life is a scavenger hunt.” It wouldn’t be fun if you knew where everything belonged. If everything just snapped into place.

Instead of starting with airtight beliefs, go trick-or-treating. The questions are your plastic pumpkin and the world’s wonders are the candy. Surprises will taste sweet if you are looking to grow, and bitter if you are afraid.

Like Halloween, you choose who you want to be.

Money Angle

Personal portfolio update: We closed on the sale of the Texas property we bought in Summer of 2021 this past week. The sale took a long time because showing it was a bit of a challenge with tenants. We asked an aspirational price in the Spring, probably 6 weeks too late to catch the insanity but still got a bid through the asking price. Unfortunately, the stock market dropped and 4 days later the buyer pulled out. We cut the home price 10%, and got into another contract quickly, but then the closing took 2 months. Alas, it’s done. We put about 40% of the proceeds into homebuilders and a world ETF and the balance in 4% t-bills. We will have a chat with our accountant this week about end-of-year tax management (selling investments that have gotten crushed in taxable accounts to offset gains on the property. Should be educational.)

Moving on…

Investing is hard. It’s a game in a complex environment. It’s hard to tell a bad decision from a good decision just based on the outcome because it’s a low signal vocation.

That said, I’ve harbored some suspicion that analytically-weak investment managers would find it convenient to hide behind those concessions. Synthesizing decisions in an open environment is much harder than solving a problem set. But what if you can’t even solve a problem set? Am I supposed to believe you can do all the hard stuff, but just choked on the solvable stuff?

This isn’t the first time I’ve wondered this. See Can Your Manager Solve Betting Games With Known Solutions?

Today’s post is in a similar vein.

Bet Sizing Is Not Intuitive (8 min read)

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 the 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 Chicago 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.”

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. Never mind 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.


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.

Last Call

Trevor Noah is leaving the Daily Show. December 8th is his last episode. Noah is one of my favorite observers of humanity.

I know who I want to take the baton.

Comedian Mo Amer.

I’ve been watching his show Mo on Netflix but just watched his 2018 stand-up special Vagabonding. It reminded me of Noah’s Afraid Of The Dark special.

I have a weak spot for comedians that can do voices (and for that matter, I also love it when animals are anthropomorphized with voices. Think any talking animal in a Super Bowl ad. It’s catnip to me.)

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