This post makes some good points, too strongly. I will now proceed to make some good points, too rambly.
F=ma is a fantastically useful concept, even if in practice the scenario is always more complicated than just “apply this formula and you win”. It’s short and intuitive and ports the right ideas about energy and mass and stuff into your brain.
Maximizing expected value is a fantastically useful concept.
Maximizing expected value at a time t after many repeated choices is a fantastically useful concept.
(Edit: see comments, there are false statements incoming. Leaving it all up for posterity.) It turns out that maximizing expected magnitude of value at every choice is very close to the optimal way to maximize your expected value at a time t after many repeated choices. Kelly is a nice, intuitive, easy heuristic for doing so.
So yes, what you really want to do is maximize something like “your total wealth at times t0 and t1 and t2 and t3 and… weighted by how much you care about having wealth at those times, or something” and the way to do that is to implement a function which knows there will be choices in the future and remembers to take into account now, on this choice having the power to maximally exploit those future choices, aka think about repeated bets. But also the simple general way to do something very close to that maximization is just “maximize magnitude of wealth”, and the intuitions you get from thinking “maximize magnitude of wealth” are more easily ported than the intuitions you get from thinking “I have a universe of future decisions with some distribution and I will think about all of them and determine the slightly-different-from-maximize-log-wealth way to maximize my finnicky weighted average over future selves’ wealth”.
Do you need to think about whether you should use your naive estimate of the probability you win this bet, when plugging into Kelly to get an idea of what to do? Absolutely. If you’re not sure of the ball’s starting location, substituting a point estimate with wide variance into the calculation which eventually feeds into F=ma will do bad things and you should figure out what those things are. But starting from F=ma is still the nice, simple concept which will be super helpful to your brain.
Kelly is about one frictionless sphere approximation to [the frictionless sphere approximation which is maximizing magnitude of wealth]. Bits that were rubbed off to form the spheres involve repeated bets, for sure.
Maximizing expected value is a fantastically useful concept.
Maximizing expected value at a time t after many repeated choices is a fantastically useful concept.
It turns out that maximizing expected magnitude of value at every choice is very close to the optimal way to maximize your expected value at a time t after many repeated choices. Kelly is a nice, intuitive, easy heuristic for doing so.
I’m not 100% sure what mathematical fact you are trying to refer to here, but I am worried that you are stating a falsehood.
Slightly editing stuff from another comment of mine:
In a one-step scenario, the Bayesian wants to maximize E[u(S⋅x)] where x is your starting money and S is a random variable for the payoff-per-dollar of your strategy. In a two-step scenario, the Bayesian wants to maximize E[u(S1⋅S2⋅x)]. And so on. If u(x)=x, this allows us to push the expectation inwards; E[u(S1⋅S2⋅x)]=E[S2⋅S1⋅x]=E[S1]⋅E[S2]⋅x (the last step holds because we assume the random variables are independent). So in that case, we could just choose the best one-step strategy and apply it at each time-step.
In other words, starting with the idea of raw expectation maximization (maximizing money, so u(x)=x, where x is our bankroll) and adding the idea of iteration, we don’t get any closer to Kelly. Kelly isn’t an approximately good strategy for the version of the game with a lot of iterations. The very same greedy one-step strategy remains optimal forever.
But I could be misunderstanding the point you were trying to communicate.
You’re absolutely right! I think I have a true intuition I’m trying to communicate, and will continue to think about it and see, but it might turn out that the entirety of the intuition can be summarized as “actually the utility is nonlinear in money”.
This post makes some good points, too strongly. I will now proceed to make some good points, too rambly.
F=ma is a fantastically useful concept, even if in practice the scenario is always more complicated than just “apply this formula and you win”. It’s short and intuitive and ports the right ideas about energy and mass and stuff into your brain.
Maximizing expected value is a fantastically useful concept.
Maximizing expected value at a time t after many repeated choices is a fantastically useful concept.
(Edit: see comments, there are false statements incoming. Leaving it all up for posterity.) It turns out that maximizing expected magnitude of value at every choice is very close to the optimal way to maximize your expected value at a time t after many repeated choices. Kelly is a nice, intuitive, easy heuristic for doing so.
So yes, what you really want to do is maximize something like “your total wealth at times t0 and t1 and t2 and t3 and… weighted by how much you care about having wealth at those times, or something” and the way to do that is to implement a function which knows there will be choices in the future and remembers to take into account now, on this choice having the power to maximally exploit those future choices, aka think about repeated bets. But also the simple general way to do something very close to that maximization is just “maximize magnitude of wealth”, and the intuitions you get from thinking “maximize magnitude of wealth” are more easily ported than the intuitions you get from thinking “I have a universe of future decisions with some distribution and I will think about all of them and determine the slightly-different-from-maximize-log-wealth way to maximize my finnicky weighted average over future selves’ wealth”.
Do you need to think about whether you should use your naive estimate of the probability you win this bet, when plugging into Kelly to get an idea of what to do? Absolutely. If you’re not sure of the ball’s starting location, substituting a point estimate with wide variance into the calculation which eventually feeds into F=ma will do bad things and you should figure out what those things are. But starting from F=ma is still the nice, simple concept which will be super helpful to your brain.
Kelly is about one frictionless sphere approximation to [the frictionless sphere approximation which is maximizing magnitude of wealth]. Bits that were rubbed off to form the spheres involve repeated bets, for sure.
I’m not 100% sure what mathematical fact you are trying to refer to here, but I am worried that you are stating a falsehood.
Slightly editing stuff from another comment of mine:
In other words, starting with the idea of raw expectation maximization (maximizing money, so u(x)=x, where x is our bankroll) and adding the idea of iteration, we don’t get any closer to Kelly. Kelly isn’t an approximately good strategy for the version of the game with a lot of iterations. The very same greedy one-step strategy remains optimal forever.
But I could be misunderstanding the point you were trying to communicate.
You’re absolutely right! I think I have a true intuition I’m trying to communicate, and will continue to think about it and see, but it might turn out that the entirety of the intuition can be summarized as “actually the utility is nonlinear in money”.