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.
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: