This is similar to the answer I got from o1-preview in ChatGPT when I originally asked with OP’s post as the text, so that’s pleasant to see. (I didn’t post anything here because I was unsure and wasn’t checking it in enough detail to repost, and so didn’t believe in publishing it without being able to improve it.)
I thought there might be some relationship at first with an appropriate transformation, but when I recalled how Kelly requires both edge and net worth, and the problem of frequency of payoffs, I lost my confidence that there would be any simple elegant relationship beyond a simple ‘more information = more returns’. Why indeed would you expect 1 bit of information to be equally valuable for maximizing expected log growth in eg. both a 50:50 shot and a 1,000,000,000:1 shot? Or for a billionaire vs a bankrupt? (Another way to think of it: suppose you have 1 bit of information on both over the market and you earn the same amount. How many trades would it take before your more informed trade ever made a difference? In the first case, you quickly start earning a return and can compound that immediately; in the second case, you might live a hundred lives without ever once seeing a payoff.)
This is similar to the answer I got from
o1-previewin ChatGPT when I originally asked with OP’s post as the text, so that’s pleasant to see. (I didn’t post anything here because I was unsure and wasn’t checking it in enough detail to repost, and so didn’t believe in publishing it without being able to improve it.)I thought there might be some relationship at first with an appropriate transformation, but when I recalled how Kelly requires both edge and net worth, and the problem of frequency of payoffs, I lost my confidence that there would be any simple elegant relationship beyond a simple ‘more information = more returns’. Why indeed would you expect 1 bit of information to be equally valuable for maximizing expected log growth in eg. both a 50:50 shot and a 1,000,000,000:1 shot? Or for a billionaire vs a bankrupt? (Another way to think of it: suppose you have 1 bit of information on both over the market and you earn the same amount. How many trades would it take before your more informed trade ever made a difference? In the first case, you quickly start earning a return and can compound that immediately; in the second case, you might live a hundred lives without ever once seeing a payoff.)