If your immediate thought is “but why consider things on a log scale in the first place?”, then remember that we’re talking about mode/order statistics, so monotonic transformations are totally fine.
Riiight, but, “totally fine” doesn’t here mean “Kelly approximately maximizes the mode as opposed to maximizing an approximate mode”, does it?
I have approximately two problems with this:
Bounding the ratio of log wealth compared to a true mode-maximizer would be reassuring if my utility was doubly logarithmic. But if it’s approximately logarithmic, this is little comfort.
But are we even bounding the ratio of log wealth to a true mode-maximizer? As I mentioned, I’m not sure a mode-maximizer is even a constant-fraction strategy.
Actually, you’re right, I goofed. Monotonic increasing transformation respects median or order statistics, so e.g. max median F(u) = F(max median u) (since F commutes with both max and median), but mode will have an additional term contributed by any nonlinear transformation of a continuous distribution. (It will still work for discrete distributions—i.e. max mode F(u) = F(max mode u) for u discrete, and in that case F doesn’t even have to be monotonic.)
So I guess the argument for median is roughly: we have some true optimum policy θ∗ and Kelly policy θK, and medianP[u|θK]F(u)≈medianP[u|θ∗]F(u), which implies medianP[u|θK]u≈medianP[u|θ∗]u as long as F is continuous and strictly increasing.
Riiight, but, “totally fine” doesn’t here mean “Kelly approximately maximizes the mode as opposed to maximizing an approximate mode”, does it?
I have approximately two problems with this:
Bounding the ratio of log wealth compared to a true mode-maximizer would be reassuring if my utility was doubly logarithmic. But if it’s approximately logarithmic, this is little comfort.
But are we even bounding the ratio of log wealth to a true mode-maximizer? As I mentioned, I’m not sure a mode-maximizer is even a constant-fraction strategy.
Sorry if I’m being dense, here.
Actually, you’re right, I goofed. Monotonic increasing transformation respects median or order statistics, so e.g. max median F(u) = F(max median u) (since F commutes with both max and median), but mode will have an additional term contributed by any nonlinear transformation of a continuous distribution. (It will still work for discrete distributions—i.e. max mode F(u) = F(max mode u) for u discrete, and in that case F doesn’t even have to be monotonic.)
So I guess the argument for median is roughly: we have some true optimum policy θ∗ and Kelly policy θK, and medianP[u|θK]F(u)≈medianP[u|θ∗]F(u), which implies medianP[u|θK]u≈medianP[u|θ∗]u as long as F is continuous and strictly increasing.