I wonder if you can recover Kelly from linear utility in money, plus a number of rounds unknown to you and chosen probabilistically from a distribution.
No, it’s fairly straightforward to see this won’t work
Let N be the random variable denoting the number of rounds. Let x = p*w+(1-p)*l where p is probability of winning and w=1-f+o*f, l=1-f the amounts we win or lose betting a fraction f of our wealth.
Then the value we care about is E[x^N], which is the moment generating function of X evaluated at log(x). Since our mgf is increasing as a function of x, we want to maximise x. ie our linear utility doesn’t change
I wonder if you can recover Kelly from linear utility in money, plus a number of rounds unknown to you and chosen probabilistically from a distribution.
No, it’s fairly straightforward to see this won’t work
Let N be the random variable denoting the number of rounds. Let x = p*w+(1-p)*l where p is probability of winning and w=1-f+o*f, l=1-f the amounts we win or lose betting a fraction f of our wealth.
Then the value we care about is E[x^N], which is the moment generating function of X evaluated at log(x). Since our mgf is increasing as a function of x, we want to maximise x. ie our linear utility doesn’t change