Comment by kqr
4 years ago
Kelly goes beyond binary outcomes. The underlying principle is the same, though: you maximise expected logarithmic wealth.
To do that you need the joint distribution of outcomes (what are the possible future scenarios and how likely are they?) Estimating this well is the trick to successful application of the Kelly criterion.
Suppose we have 100 sequential bets with distribution U(-1,1.1) on each. How would we apply Kelly here?
You wouldn't unless you could vary your exposure to such a sequential bet.
Suppose you can though. For simplicity, suppose you can expose yourself to 0.4U(-1, 1.1), 40U(-1, 1.1), or any other fractional amount F U(-1, 1.1) you might like. Kelly is a technique for choosing F (maybe you had some other idea in mind like that you have to buy into a bet on U(0, 2.1) -- if so, that's nearly equivalent other than putting bounds on F -- the idea of maximizing expected logarithm will carry through to other bet structures).
Going through the motions, suppose you're starting with a bankroll B then you want to choose some ratio F=rB maximizing the expected logarithm of the bet. The distribution of your outcome is another uniform distribution U(B-rB, B+1.1rB), and you want to choose r maximizing the expected logarithm of that distribution. The details of that are probably beyond the scope of a HN comment, but you wind up with r approximately equal to 0.13624.
If you'd like you could plot the result of many instances of 100 such sequential bets with r varying. You'll find that those with r around 0.13624 will usually be much larger than for other choices of r.