Maximizing expected value is a fantastically useful concept.
Maximizing expected value at a time t after many repeated choices is a fantastically useful concept.
It turns out that maximizing expected magnitude of value at every choice is very close to the optimal way to maximize your expected value at a time t after many repeated choices. Kelly is a nice, intuitive, easy heuristic for doing so.
I’m not 100% sure what mathematical fact you are trying to refer to here, but I am worried that you are stating a falsehood.
Slightly editing stuff from another comment of mine:
In a one-step scenario, the Bayesian wants to maximize E[u(S⋅x)] where x is your starting money and S is a random variable for the payoff-per-dollar of your strategy. In a two-step scenario, the Bayesian wants to maximize E[u(S1⋅S2⋅x)]. And so on. If u(x)=x, this allows us to push the expectation inwards; E[u(S1⋅S2⋅x)]=E[S2⋅S1⋅x]=E[S1]⋅E[S2]⋅x (the last step holds because we assume the random variables are independent). So in that case, we could just choose the best one-step strategy and apply it at each time-step.
In other words, starting with the idea of raw expectation maximization (maximizing money, so u(x)=x, where x is our bankroll) and adding the idea of iteration, we don’t get any closer to Kelly. Kelly isn’t an approximately good strategy for the version of the game with a lot of iterations. The very same greedy one-step strategy remains optimal forever.
But I could be misunderstanding the point you were trying to communicate.
You’re absolutely right! I think I have a true intuition I’m trying to communicate, and will continue to think about it and see, but it might turn out that the entirety of the intuition can be summarized as “actually the utility is nonlinear in money”.
I’m not 100% sure what mathematical fact you are trying to refer to here, but I am worried that you are stating a falsehood.
Slightly editing stuff from another comment of mine:
In other words, starting with the idea of raw expectation maximization (maximizing money, so u(x)=x, where x is our bankroll) and adding the idea of iteration, we don’t get any closer to Kelly. Kelly isn’t an approximately good strategy for the version of the game with a lot of iterations. The very same greedy one-step strategy remains optimal forever.
But I could be misunderstanding the point you were trying to communicate.
You’re absolutely right! I think I have a true intuition I’m trying to communicate, and will continue to think about it and see, but it might turn out that the entirety of the intuition can be summarized as “actually the utility is nonlinear in money”.