Some Questions on Utility Funcitons

From Stanford Wong's BJ21

Posted by MathProf on 15 Dec 1997, at 4:20 a.m., in response to Utility functions and optimal f , posted by Karel Janecek on 14 Dec 1997, at 10:29 a.m.

A couple of questions regarding the above post:

A class of appropriate utility functions is: U(x) = ((x+1)^a-1)/a, where a is a parameter in the interval (-infinity;1). a=1 is not appropriate any more since we get the linear utility function (betting everything) which does not posses some of the properties required on a utility function. If we take the limit a-->0, we get exactly the logarithmic utility function.

What properties are we concerned with. Linear Utility is not Risk-Averse, but Risk-Neutral, and if a person follows it that will go Bankrupt almost surely (with Probability 1). But does that preclude it from being a utility fct?

There are other very interesting question stemming from this: for example, there will probably be some border-line value A in the interval (0;1) where the utility function is "good" for a lower than A, and "bad" for a greater than or equal to A in the sence hat a = A and greater will give optimal f for which
the player's bankroll will converge to zero with robability one. Unfortunately, the calculations become very complicated.

I would attempt this in the following way. For these other utility fcts, isn;t there a second-order approximation, which then involves E(X) and E(X^2), just as in log-utility? So we can compute a "k" which leads to a betting fraction f=k*E(X)/E(X^2). Now if this is over "2", the utility causes us to bet double-kelly, which will lead to bankruptcy. And if we do less, it will not? [I think this happens with the sqrt fct (alpha=2) ]

Or have missed some important point here?