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.
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.
You do, however, maximise your long-term growth rate, regardless of any utility function. If you consistently overbet Kelly[1], you will consistently impoverish yourself, even while your “expected” utility is skyrocketing, but confined to an increasingly tiny sliver of probability space. In the limit, the probability of being in profit goes to zero. The “expectation” in this situation is the oppposite of what you can expect to see.
You do, however, maximise your long-term growth rate, regardless of any utility function.
When you say “growth rate” you’re picking out one specific metric from an infintie set of other reasonable choices!
Of course if you focus on ratios, you tautologically end up maximising the logarithm!
But you’re still doing a lot of normative work when you make the “rate of growth”, defined as , the ideal you seek to maximise the expected value of!
If you run a bunch of simulations exploring the results achieved by different betting strategies and then compare the average results using the geometric mean then yes, Kelly betting wins.
If you plot some simulations on a chart and make your y-axis logarithmic then yes, the chart seems to show how Kelly beats everything almost always.But by doing the analysis this way you’ve already baked in the conclusion.
When we start to talk along the lines of “Yeah this maximises expected utility—but” this is a sign that there’s a type error somewhere. There is no but—your utility function is definitionally the thing you want to maximise the expected value of.
Right, maximizing median wealth is also the same as ‘maximize the chance that I have the most bankroll to spare for any better betting opportunities that come along in the future’ afaik
maximize the chance that I have the most bankroll to spare for any better betting opportunities that come along in the future
What does “most” mean? If you start with W and go all-in on the coin flip game—you end up with 2W with probability
2W is the “most” you can possibly end up with to spare when the next betting opportunity comes along.
So by that framing, going all-in is what maximises the chance you have the most bankroll to spare.
(I’m not pretending that is a good argument I just made—I’m just pointing out that these desiderata we’re trying to express in natural language have lots of room for interpretation when it comes to turning them into math—and the only sensible way to resolve this ambiguitiy is to start with your utility function and derive the risk taking policy from that, not the other way around!)
Maybe it would help if you construct an explicit concrete model? You’re welcome to define what future opportunities will come along after this bet (or even a distribution of possible future opportunities)
Are you claiming that after you build this concrete model—Kelly betting will emerge as the objectively optimal strategy regardless of the agent’s preferences and regardless of whether we add a safety net/income stream into the picture?
Or are you making a softer (seemingly irrelevant) claim about what happens with geometric means/average growth rates when we don’t account for safety nets and income streams?
While the outcomes that drive up the expected dollars are mainly in the regime where your utility is linear in dollars, you should bet crazily; after that point, you should be hedging more; Kelly works (uniquely?) well in the long-run; but in the real world you could know what utility regime you’re in / how long of a run you’re in or something like that.
I think log utility mischaracterizes people’s utility wrt money function in some ways, but disagree with the reasons you give. The main departures afaict is that real utility follows mutltiple sigmoids around decision relevant amounts of money (eg having runway vs living paycheck to paycheck, minimal retirement money, large lifestyle change money) and the fact that real betting opportunities are heterogeneous rather than continuous eg since high conviction bets come along at unpredictable intervals, many people have a barbell strategy of mostly index funds plus a few higher conviction concentrated bets.
There’s also that we can’t treat reachable utility via spending money as the same between people, but that’s outside scope.
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.
You do, however, maximise your long-term growth rate, regardless of any utility function. If you consistently overbet Kelly [1] , you will consistently impoverish yourself, even while your “expected” utility is skyrocketing, but confined to an increasingly tiny sliver of probability space. In the limit, the probability of being in profit goes to zero. The “expectation” in this situation is the oppposite of what you can expect to see.
ETA: by a sufficiently large amount, a sufficiently modest overbet being merely suboptimal.
When you say “growth rate” you’re picking out one specific metric from an infintie set of other reasonable choices!
Of course if you focus on ratios, you tautologically end up maximising the logarithm!
But you’re still doing a lot of normative work when you make the “rate of growth”, defined as
If you run a bunch of simulations exploring the results achieved by different betting strategies and then compare the average results using the geometric mean then yes, Kelly betting wins.
If you plot some simulations on a chart and make your y-axis logarithmic then yes, the chart seems to show how Kelly beats everything almost always. But by doing the analysis this way you’ve already baked in the conclusion.
In an analogous way to how odds ratios are isomorphic to probabilities—but in some cases cause less confusion when we try reason about them—I think it’s way more productive to reason about your utility function than it is to talk about growth rates
When we start to talk along the lines of “Yeah this maximises expected utility—but” this is a sign that there’s a type error somewhere. There is no but—your utility function is definitionally the thing you want to maximise the expected value of.
Right, maximizing median wealth is also the same as ‘maximize the chance that I have the most bankroll to spare for any better betting opportunities that come along in the future’ afaik
What does “most” mean? If you start with W and go all-in on the coin flip game—you end up with 2W with probability
2W is the “most” you can possibly end up with to spare when the next betting opportunity comes along.
So by that framing, going all-in is what maximises the chance you have the most bankroll to spare.
(I’m not pretending that is a good argument I just made—I’m just pointing out that these desiderata we’re trying to express in natural language have lots of room for interpretation when it comes to turning them into math—and the only sensible way to resolve this ambiguitiy is to start with your utility function and derive the risk taking policy from that, not the other way around!)
unknown number of rounds before this bet expires and or other bets come along.
Maybe it would help if you construct an explicit concrete model? You’re welcome to define what future opportunities will come along after this bet (or even a distribution of possible future opportunities)
Are you claiming that after you build this concrete model—Kelly betting will emerge as the objectively optimal strategy regardless of the agent’s preferences and regardless of whether we add a safety net/income stream into the picture?
Or are you making a softer (seemingly irrelevant) claim about what happens with geometric means/average growth rates when we don’t account for safety nets and income streams?
I think there’s a middle ground statement:
I think log utility mischaracterizes people’s utility wrt money function in some ways, but disagree with the reasons you give. The main departures afaict is that real utility follows mutltiple sigmoids around decision relevant amounts of money (eg having runway vs living paycheck to paycheck, minimal retirement money, large lifestyle change money) and the fact that real betting opportunities are heterogeneous rather than continuous eg since high conviction bets come along at unpredictable intervals, many people have a barbell strategy of mostly index funds plus a few higher conviction concentrated bets.
There’s also that we can’t treat reachable utility via spending money as the same between people, but that’s outside scope.