I think there is only a fixed-proportion betting rule (i.e. “bet P% of your bankroll” for fixed P) for CRRA utility functions, because if risk aversion varies then the betting rule must also vary. But I’m not sure how to prove that.
ETA: Actually I think it shouldn’t be too hard to prove that using the definition of CRRA. You could do something like, assume a fixed-proportion betting rule exists for some constant P, and then calculate the implied relative risk aversion and show that it must be a constant.
For constant-relative-risk-aversion (CRRA) utility functions, the Kelly criterion is optimal iff you have logarithmic utility. For proof, see Samuelson (1971), The “Fallacy” of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling.
I think there is only a fixed-proportion betting rule (i.e. “bet P% of your bankroll” for fixed P) for CRRA utility functions, because if risk aversion varies then the betting rule must also vary. But I’m not sure how to prove that.
ETA: Actually I think it shouldn’t be too hard to prove that using the definition of CRRA. You could do something like, assume a fixed-proportion betting rule exists for some constant P, and then calculate the implied relative risk aversion and show that it must be a constant.