as a one time bet sure, but there are obviously bankroll considerations for iterated. kelly criterion is about growing your bankroll the fastest you can without busting so you can't take advantage of the iterated bet any more.
There are different ways to exactly define "growing your bankroll the fastest without going bust". Making that your goal and then choosing the one specific mathematical instantiation of the concept that leads to Kelly betting - is equivalent to declaring you have log utility.
If you have any utility function other than log(W) then you don't maximize your expected utility by Kelly betting - even if there's many repeated bets.
This strikes most people as being insanely agressive
I don't think so. ~Everyone can see that this is super obviously a good deal.
but this is paradoxical because the assumptions underpinning the analysis are actually wildly conservative
The bankrupt is infinitely bad assumption is implicit in the log-wealth model and definitely get some people confused when they first encountered it. It is definitely a wildly conservative assumption that is wrong, I don't trust Kelly at like $0-$100. But it feels to me that this pitfall is not as big of an issue as you're saying, because in most realistic situations 1. the utility of having low digit money won't affect the Kelly calculation much for mediocre deals (intuition; didn't math it out), 2. you want to discount on Kelly (or any modified-Kelly) anyways because of uncertainty in the probability/returns of the bets you're making.
> I don't think so. ~Everyone can see this is super obviously a good deal
There's a difference between thinking it's a good deal and being willing to bet 50% of your net worth on it. Do you think if we polled a bunch of people, almost everyone would say they'd bet at least as agressivlely as Kelly prescribes? (If you do, want to make a meta-bet about this? :P )
I pointed out the negative infinity thing not just to make fun of the singualarity at 0, but to gesture at the fact that in general we should consider our utility functions as being way less curved than a logarithm.
In similar vein to how log utility treats the difference between being flat out broke and only having $100 as infinite, wheras to you the difference is negligible - it's also wildly exagerating the difference between 100 and 1,000, and 1,000 vs 10,000.
It's not just below $100 that you souldn't trust kelly. If you have an anual salary of 100k you shoudln't trust kelly anywhere below like 500k!
You're right that the issue is that in the real world it's very easy to be wrong about the size of your edge and most of the time any major edge comes along with a capped bet size. But distinguishing between "I'm very risk averse" and "I'm not very risk averse but it's very hard to eek out big edges" is useful - and does have practical implications in some cases (going to write about this in future post!)
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1Not making a real money bet because it seems difficult to operationalize / flush out the details enough that I would think I have enough edge. People imo will correctly give a more conservative number if they think the question is realistic, and they will give closer to 50% if they think it is an idealized scenario where all they have is money. But I will say 30% of people would give >=50% if they understand the scenario as mathematical.
Also, I was saying something weaker. I disagreed on "This strikes most people as being insanely agressive". I am saying that people would, after being told the correct answer (i.e. in retrospect), think/tell you that the mathematically correct answer is not insanely aggressive. Even if 50% is higher than what they said would personally bet, I think most people would not say and would disagree that it is insanely aggressive.
1I think if you start asking people this question, even educated people, you'll be supprised!
While the scenario is idalized in the sense of you can know the payoffs and odds with certianty - there's no need to stipulate "all they have is money" - they can have a complex utility function involving a thousand inputs, as long as the only input that changes based on the bet they make is money.
In response to "Why" reacts:
Let's think about another hypothetical. Say you were forced to choose between 2 bad options:
Option 1: You lose all your wealth save for $10,000
Option 2: We flip a coin, heads you lose all your wealth save for $100 dollars, tails you lose all your wealth save for 1 million dollars (and if you currently are worth less than a million you actually recieve money to get your net worth up to this figure)
We're not leaning on the singularity at 0 here.
Log utility says these options have the same value. But obviously someone with a safety net and healthy future earning potential should way prefer option 2
Ok I understand what you're saying now. My reaction is that we should just add the expected current value of all the money you will make in the future (maybe discounted and also conditional on you making the bet), to your current wealth and then kelly as if you have that much money. This seems like a valid critique of how people Kelly bet currently, but I still disagree the correct response to this hypothetical is "we should consider our utility functions as being way less curved than a logarithm". I think people do genuinely value wealth roughly logarithmically so if you don't make any money in the future then Kelly is correct.
I do understand "if you have an anual salary of 100k you shoudln't trust kelly anywhere below like 500k" now and I agree.
I don't mind whether we frame it as utility curves being flatter than logarithmic, or as logarithmic curves but shiffted to the left - both are approximation of the real function regardless. (And mathematically I don't think there's even a difference... The slope of ln(x) is 1/x so shifting it left does make it flatter)
The high level point is that both framings seem to imply we should bet far more aggressively that how Kelly Criterion is typically applied
[More leisurely version of this post in video form here]
Imagine a wealthy, eccentric person offers to play a game with you. You flip 2 fair coins, and if either lands TAILS, you win. If both land HEADS, you lose. This person is willing to wager any amount of money you like on this game (at even-money). So whatever you stake, there's a ¼ chance you lose it and a ¾ chance you double it.
There's no doubt about the integrity of the game - no nasty tricks, it's exactly what it looks like, and whoever loses really will have to honour the bet.
How much money would you put down? It's very likely your initial answer to this question is far too low.
The Von Neumann-Morgenstern theorem says we should act as if we are maximising the expected value of some utility function - and when it comes to this decision the only meaningful variable our decision affects is how much money we have.
So to arrive at our correct bet size we just need to figure out the shape of our utility vs wealth curve.
This curve is different for everyone, but in general we can say it should be upward sloping (more money is better than less) and get less steep as we move to the right (diminishing returns of each additional dollar)
When we think about an upward sloping curve with diminishing returns, the obvious choice that comes to mind is the log. i.e.
Where
We don't have to choose the log here, (there's nothing actually special about it), but it's a reasonable place to start our analysis from. Sizing our bets to maximise the log of our wealth is also known as the Kelly Criterion
Intuitively, log utility says every doubling of money leads to same incremental increase in wellbeing (so the happiness bump going from living on 50k to 100k a year is the same as going from 100k to 200k is the same as going from 200k to 400k etc.)
This won't be exactly your preferences, but hopefully this feels "close enough" for you to be interested in the implications.
If we start with a wealth of
So our Expected utility is:
Which is maximised when
So Kelly Criterion says you should bet half of everything you have on the outcome of this coinflip game.
This strikes most people as being insanely agressive - but this is paradoxical because the assumptions underpinning the analysis are actually wildly conservative.
As your wealth approaches zero, the log goes to negative infinity. So log utility is saying that going bankrupt is not just bad, but infinitely bad (akin to being tortured for eternity).
This is a bit overdramatic - A young American doctor who just finished med school with a small amount of student debt is not "poor" in any meaningful sense, and she's certainly not experiencing infinitely negative wellbeing.
For anyone in the class of "people who might see this post" - when we compute our wealth
If you re-do the analysis but treat W as being just 20% higher due to unrealised future earnings, the optimal betting fraction according to log-utility jumps up to 60%.
Or if you think the peak of your career is still ahead of you - and model things so that your future earnings exceed your current net worth - the answer becomes bet every single cent you have on this game.
This is deeply unintuitive. And my stance is that in this idealized situation, where you really can be certain of a huge edge, it's our intuitions that are wrong.
I honestly would go fully all-in on a game like this (if anyone thinks I'm joking and has a lot of money, please try me 😉)
But don't go and start betting huge sums of money on my account just yet - in slightly more realistic settings there are forces which push us back closer to the realm of "normal" risk aversion. I plan to cover this in my next post.
Pretending for now that AI isn't about to transform the world beyond recognition...