The part about the Kelly criterion that has most attracted me is this:
That thing is that betting Kelly means that with probability 1, over time you’ll be richer than someone who isn’t betting Kelly. So if you want to achieve that, Kelly is great.
So with more notation, P(money(Kelly) > money(other)) tends to 1 as time goes to infinity (where money(policy) is the random score given by a policy).
This sounds kinda like strategic dominance—and you shouldn’t use a dominated strategy, right? So you should Kelly bet!
The error in this reasoning is the “sounds kinda like” part. “Policy A dominates policy B” is not the same claim as P(money(A) >= money(B)) = 1. These are equivalent in “nice” finite, discrete games (I think), but not in infinite settings! Modulo issues with defining infinite games, the Kelly policy does not strategically dominate all other policies. So one shouldn’t be too attracted to this property of the Kelly bet.
(Realizing this made me think “oh yeah, one shouldn’t privilege the Kelly bet as a normatively correct way of doing bets”.)
Yes, but there’s an additional thing I’d point out here, which is that at any finite timestep, Kelly does not dominate. There’s always a non-zero probability that you’ve lost every bet so far.
When you extend the limit to infinity, you run into the problem “probability zero events can’t necessarily be discounted” (though in some situations it’s fine to), which is the one you point out; but you also run into the problem “the limit of the probability distributions given by Kelly betting is not itself a probability distribution”.
The part about the Kelly criterion that has most attracted me is this:
So with more notation, P(money(Kelly) > money(other)) tends to 1 as time goes to infinity (where money(policy) is the random score given by a policy).
This sounds kinda like strategic dominance—and you shouldn’t use a dominated strategy, right? So you should Kelly bet!
The error in this reasoning is the “sounds kinda like” part. “Policy A dominates policy B” is not the same claim as P(money(A) >= money(B)) = 1. These are equivalent in “nice” finite, discrete games (I think), but not in infinite settings! Modulo issues with defining infinite games, the Kelly policy does not strategically dominate all other policies. So one shouldn’t be too attracted to this property of the Kelly bet.
(Realizing this made me think “oh yeah, one shouldn’t privilege the Kelly bet as a normatively correct way of doing bets”.)
Yes, but there’s an additional thing I’d point out here, which is that at any finite timestep, Kelly does not dominate. There’s always a non-zero probability that you’ve lost every bet so far.
When you extend the limit to infinity, you run into the problem “probability zero events can’t necessarily be discounted” (though in some situations it’s fine to), which is the one you point out; but you also run into the problem “the limit of the probability distributions given by Kelly betting is not itself a probability distribution”.