Kelly Bet on Everything
Cross-posted, as always, from Putanumonit.
It’s a core staple of Putanumonit to apply ideas from math and finance out of context to your everyday life. Finance is about making bets, but so is everything else. And one of the most useful concepts related to making bets is the Kelly criterion.
It states that when facing a series of profitable bets, your wagers should grow proportionally with your bankroll and with your edge on each bet. Specifically, that you should bet a percentage of your bankroll equivalent to your expected edge — if a bet has a 55% chance to go your way your edge is 55%-45%=10% and you should risk 10% of your bankroll on it (assuming equal amounts wagered and won). There could be reasons to avoid betting the full Kelly in practice: you’re not sure what your edge is, your bet size is limited, etc. But it’s a good guide nevertheless.
People’s intuition is usually that Kelly bets are too aggressive, that betting half of everything you have a on 75%-25% bet is too wild. But the Kelly criterion is actually quite conservative in that it maximizes not the expected size of your bankroll but it’s expected logarithm. “Exponential” means “fast and crazy”; logarithm is the inverse of that. It’s slow and cautious. If you have $1,000 and you’d risk no more than $750 for an equal chance to win $3,000, you’re logarithmic in dollars and should “bet the Kelly”.
Log scales apply to the difficulty and value you get for most things. Life satisfaction grows with log(money). Making a new friend is probably one tenth as valuable to someone who has 10 friends than to someone who has one, so your social life depends on log(friends). It’s equally hard to double one’s number of blog readers, sexual partners, job offers etc regardless of how many you have, as opposed to incrementing each by a fixed amount. It’s equally valuable too.
And so, for most things, it makes sense to bet the Kelly. You’ll need to find out what bets are available, where your edge is, and what your bankroll is.
Money
Let’s start with the obvious one. What kind of Kelly bets can you make with money? Investments are the obvious one, and standard investment advice is to switch to high-risk-high-return assets when you have some money to spare.
You can also make bets on your ability to make money: take on a side project, look for a new job, start your own business, ask for a raise. Each one entails a risk and a possible reward. Your bankroll is your literal bankroll, your edge is your ability to make money for yourself or your employer.
People have a tendency to think that if they’re paid $N a month their value to their employer is something like N and half, but that often way off. Some people are worth less than what they are paid, but are kept around because their boss can’t tell. Some people are worth 10x their salary — an employer has no reason to pay you more if you don’t ask for it. I quit a job once and immediately got offered a 30% raise to come back. I did some math on what I’m worth, gambled on asking for 50%, and got it.
Friends
When your friendships are few and tenuous, people’s inclination is to play it safe and conform to the crowd. It won’t make you a social star, but it won’t turn people away either. But if you have an edge in popularity and enough close friends to fall back on you can make some bets on your own vision.
When I was younger and struggled to make friends I’d just wait to be invited to parties. When I finally figured it out and acquired a rich social life I started throwing my own events the way I like them: controversial topic parties, naked retreats in the woods, psychedelic rationality workshops. Each one is a gamble — the event could fail or people could just not show up. In either case I’d lose some of the status and goodwill that allowed me to plan those events in the first place. But when it works the payoff is equally great.
Creative Talent
Whatever creative outlet you have, you get better by getting feedback from the audience. Show people your paintings, read them your poems, invite them to your shows, link them to your blog. This is a gamble — if people don’t like what you’re making you won’t get their attention next time.
When I just arrived in NYC I was doing stand-up and would perform at bringer shows where you get stage time if you bring 3 or 4 paying guests. My ability to do that depended on the number of friends willing to humor me (bankroll) and my humor (edge). By the time I got decent enough to get an invite to a non-bringer show I had just about run out of comedy-tolerating friends to call on.
Romance
The most obvious way to bet on yourself in romance is to flirt with people “outside of your league”, your bankroll being in part your ability take rejection in stride and stay single for longer. The same applies the other way, with making the bet on breaking up a relationship that is merely OK in hopes of something better.
But you can also bet on an existing relationship. If the person you’re dating just got into a school or job in a faraway city your ability to go long-distance for a while depends a lot on the bankroll of relationship security you have. Ethical non-monogamy is a similar gamble: if you’re don’t have an edge in making your partner happy they may leave you. If you do, their happiness only doubles for their ability to date other people, and polyamory makes you all the more attractive as a partner.
Polyamory makes bad relationships worse and good ones better; if you only know people who opened up when their relationship started deteriorating you’re liable to miss this fact.
Sanity
Psychedelics can drive you insane. They can also make you saner than you’ve every been. The same applies to meditation, mysticism, esoteric ideologies, and whatever else Bay Area Rationalists are up to. Epistemic Rationality is your bankroll and your edge.
Reputation
A lot of people are seeing the rise in callout and cancel culture purely as a threat, a reason to go anonymous, lock their accounts, hide in the dark forest of private channels. But where there’s threat there’s also opportunity, and where reputations can be lost they can also be made. Chaos is a ladder.
In 2015 Scott Aaronson’s blog comment went viral and threatened to spark an outrage mob. Aaronson didn’t expect that popular feminist writers would dedicate dozens of pages to calling him an entitled privileged asshole for expression his frustrations with dating as a young nerd. But he also didn’t expect that Scott Alexander would write his most-read blog post of all time in defense of Aaronson, and that the entire Rationalist community would mobilize behind him. This wouldn’t have happened if Aaronson hadn’t proven himself a decent and honest person, writing sensitively about important topics under his real name. Aaronson’s reputation both online and in his career only flourished since.
Children
Having children is a bet that you have enough of an edge on life that you can take care of another human and still do well. The payoff is equally life-changing.
Risk Averse Irrationalists
I wrote this post because of my endless frustration with my friends who have the most slack in life also being the most risk averse. They have plenty of savings but stay in soul-sucking jobs for years. They complain about the monotony of social life but refuse to instigate a change. They don’t travel, don’t do drugs, don’t pick fights, don’t flirt, don’t express themselves. They don’t want to think about kids because their lives are just so comfortable and why would you mess with that?
They often credit their modest success to their risk-aversion, when it’s entirely due to them being two standard deviations smarter than everyone they grew up with. By refusing to bet on themselves they’re consigned forever to do 20% better than the most average of their peers. To make 20% more money with a 20% nicer boyfriend and 1.2 Twitter followers.
And partly, I wrote this post for me. I spent my twenties making large bets on myself. I moved to the US nine years ago today, all alone and with a net worth of $0. I found polyamory and the love of my life. I started a blog under my real name, with my real opinions, on real topics.
Now in my mid-thirties my life is comfortable, my slack is growing, and I’m surrounded by younger friends who know all about discretion and little about valor. This post is a reminder to keep looking for my edge and keep pushing the chips in. There’s plenty more to be won.
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On the whole I agree with Raemon’s review, particularly the first paragraph.
A further thing I would want to add (which would be relatively easy to fix) is that the description and math of the Kelly criterion is misleading / wrong.
The post states that you should:
However the correct rule is:
(emphasis added)
The 2 definitions give the same results for 1:1 bets but will give strongly diverging results with other odds.
In addition the post gives a method of calculating one’s edge which gives correct results for 1:1 bets. It is not entirely clear how one would use the formula for non 1:1 bets but it doesn’t seem to indicate a calculation which would give one’s edge correctly. (55%-45%=10% doesn’t seem to readily generalise correctly in the same way that (55%-50%)/50%=10% does).
The post never states that the definition is only for 1:1 bets so the impression given by the post is that these formulae can be used in the cases given in the post. However there are no guarantees that any of the examples in the post are 1:1 bets.
(The post does mention that the example is based on 1:1 bets but it doesn’t imply that the calculation as given only works for such bets.)
As a result the post effectively ends up recommending making incorrectly sized bets. For non 1:1 bets it is possible to calculate recommended bet sizes which have an expected negative impact on the logarithm of ones bankroll and as such actively makes one’s life worse.
(e.g. the bet odds are at 25% but you believe there’s a 50% chance that it will resolve true then your edge is 100% (calculated properly, not using the method in the post as I’m not sure how to use that) and the post would suggest betting your entire bankroll. The correct Kelly calculation gives 1⁄3rd of your bankroll)
Without this correction I would strongly recommend against including this post in the review.
This is a useful clarification. I use “edge” normally to include both the difference in probability of winning and losing and the different payout ratios. I think this usage is intuitive: if you’re betting 5:1 on rolls of a six-sided die, no one would say they have a 66.7% “edge” in guessing that a particular number will NOT come up 5⁄6 of the time — it’s clear that the payout ratio offsets the probability ratio.
Anyway, I don’t want to clunk up the explanation so I just added a link to the precise formula on Wikipedia. If this essay gets selected on condition that I clarify the math, I’ll make whatever edits are needed.
So there’s a technical definition of edge which is your expected gain for every unit that you bet, given your own probability and the bet odds.
I agree that not clumping up the post is probably best but to make the post correct I suggest adding the underlined text into the definition in case people don’t click the link.
On one hand, AFAICT the math here is pretty fuzzy, and one could have written this post without it, instead just using the same examples to say “you should probably be less risk averse.” I think, in practice for most people, the math is a vague tribal signifier that you can trust the post, to help the advice go down.
But, I see this post in a similar reference class to Bayes’ Theorem. I think most people don’t actually need to know Bayes Theorem. They need to remember a few useful heuristics like “remember the base rates, not just the salient evidence you came across”, and “think about relative likelihood ratios between multiple hypothesis.”
Nonetheless, an important aspect of The LessWrong Project™ is building a unified worldmap, looking to understand the lawfulness that underlies our heuristics, and when our cleanest laws don’t quite apply to our messy situations, keeping an eye out for how we could someday later better integrate everything into a single unified model.
My understanding of the Kelly Criterion makes it something like “idealized Bayes Theorem but for instrumental rationality instead of epistemic.”
I’d kinda like there to be a version of this post that’s more rigorous, that attempts to take some of Jacob’s fuzzy examples and do more math to them (maybe just a couple, to fully illustrate the idea). I confess this is the sort of thing I don’t expect to really be directly helpful to anyone – probably most people read this post, absorb the general lesson of “you should be less risk averse” and move on with their day. But it’s nice to have more details available when you peer under the hood.