Kelly optimizes Logan’s utility function, and NOT Jane’s.
Kelly doesn’t optimize either of those things. When offered the bets that ruin Linda, we see that she doesn’t optimize Linda’s utility function (she bets like Logan in that situation); and when offered the bets that ruin Logan, we see that she doesn’t optimize Logan’s utility function (this is explored in the final section). A large part of the point of the previous post is that Kelly betting isn’t about optimizing a utility function.
this is simply incorrect
I’m not sure what you think is incorrect. I assume you don’t mean I’m wrong about how I use the term. I guess you mean “no, the strategy that you describe as betting Kelly does not have that effect in this situation”? (And I assume by that strategy, you’re thinking of the fractional-betting thing, with unlimited subdivisions allowed?)
I also guess you misunderstand what I mean by rank-optimizing. I gave a technical definition in the linked post as
A strategy λ is rank-optimal if for all strategies μ,
limn→∞P(Vn(λ)≥Vn(μ))=1.
(And we can also talk about a strategy being “equally rank-optimal” as or “more rank-optimal” than another, in the obvious ways. I’m pretty sure this will be a partial order in general, and I suspect a total order among strategy spaces we care about.)
And it seems clear to me that under this definition, fractional betting (with unlimited subdivisions) is indeed more rank-optimal than betting everything every time.
Perhaps my non-technical definition made you think the technical definition was something else? Maybe “with probability tending to 1” would have been clearer.
Yes, my objection is solved with “probability tending to 1”. At any finite point, the probability is less than 1, and the magnitude of win in those cases tends to infinity.
By Jane, do you mean Linda?
Kelly doesn’t optimize either of those things. When offered the bets that ruin Linda, we see that she doesn’t optimize Linda’s utility function (she bets like Logan in that situation); and when offered the bets that ruin Logan, we see that she doesn’t optimize Logan’s utility function (this is explored in the final section). A large part of the point of the previous post is that Kelly betting isn’t about optimizing a utility function.
I’m not sure what you think is incorrect. I assume you don’t mean I’m wrong about how I use the term. I guess you mean “no, the strategy that you describe as betting Kelly does not have that effect in this situation”? (And I assume by that strategy, you’re thinking of the fractional-betting thing, with unlimited subdivisions allowed?)
I also guess you misunderstand what I mean by rank-optimizing. I gave a technical definition in the linked post as
And it seems clear to me that under this definition, fractional betting (with unlimited subdivisions) is indeed more rank-optimal than betting everything every time.
Perhaps my non-technical definition made you think the technical definition was something else? Maybe “with probability tending to 1” would have been clearer.
Yes, my objection is solved with “probability tending to 1”. At any finite point, the probability is less than 1, and the magnitude of win in those cases tends to infinity.