Thank you for spending time to write such a long response to my questions. I do have the following comments: Perhaps the three utility functions, instead of describing the risk preference of the better, could instead describe the three different types of system the better bets on? 1. The concave utility function represents a betting system with lots of randomness like stocks/options/blackjack... Because the outcome is governed by randomness, if you bet too big, you are always at the risk of losing it all due to the random nature of the outcome. So, you bet less than optimum? 2. The linear utility function represents a betting system with non random payoff where input and output are known and definitive. If your bet is selling widgets, your payoff depends solely on the number you sell. So, you bet the farm. 3. The convex function represent a betting system where the more you bet the higher your payoff. If your bet is like starting a social media business where the more people you sign up the more your return is. In that case, you want to bet more and more on it as the payoff escalates. So you bet more than the farm, and then some (e.g. starting and running Uber and Lyft).
You're welcome. Regarding your examples, think of utility (e.g. Kelly) as _your_ risk preference. The examples you've described are _your_ risk preferences for the 3 cases. A typical casino game fits with risk-affine utility. You already know the house has an edge but that's not stopping you from going there. Your example of a social media business is an interesting one. That's consistent with VC firms making subsequent investments in the social media firm at ever-higher valuations. But VC firms definitely seek to earn a risk premium. At the very least, the VC likely has a risk budget constraint.
"Increasing the risk aversion parameter (smaller Kelly fraction) would reduce the curvature of the risk-averse utility curve." My apologies, this sentence is incorrect. This would _increase_ the curvature, not reduce. That is, it would increase the concavity.
I think your original statement is more correct. With < 1/2 Kelly you stays at the more linear, less concave part of the curve:
My choice of words wasn't helping. Sorry, I've made this more confusing than it needs to be. Let's start with an image. Looking at the previous plot labeled 'risk-averse', think of the bottom right part of the utility curve as anchored at the origin and the upper right part being bent downward. This represents an increase in curvature and an increase in risk aversion. Now, let's formalize this a bit: The coefficient of relative risk aversion (RRA) is defined as - u"(w)/u'(w). The Kelly utility function is a log utility function E[log(w)]. Fortumately, the RRA for log utility is rather simple, 1/w. We call this 'fractional Kelly'. Now you know where that Kelly fraction comes from