Well put. I agree that we should try to maximize the value that we expect to have after playing the game.
My claim here is that just because a statistic is named “expected value” doesn’t mean it’s accurately representing what we expect to happen in all types of situations. In Alice’s game, which is ergodic, traditional ensemble-averaging based expected value is highly accurate. The more tickets Alice buys, the more her actual value converges to the expected value.
In Bob’s game, which is non-ergodic, ensemble-based expected value is a poor statistic. It doesn’t actually predict the value that he would have. There’s no convergence between Bob’s value and “expected value”, so it seems strange to say that Bob “expects to get” the result of the ensemble average here.
You can certainly calculate Bob’s ensemble average, and it will have a higher result than the temporal average (as I state in my post). My claim is that this doesn’t help you, because it’s not representative of Bob’s game at all. In those situations, maximizing temporal average is the best you can do in reality, and the Kelly criterion maximizes that. Trying to maximize ensemble-based expected value here will wipe you out.
Well put. I agree that we should try to maximize the value that we expect to have after playing the game.
My claim here is that just because a statistic is named “expected value” doesn’t mean it’s accurately representing what we expect to happen in all types of situations. In Alice’s game, which is ergodic, traditional ensemble-averaging based expected value is highly accurate. The more tickets Alice buys, the more her actual value converges to the expected value.
In Bob’s game, which is non-ergodic, ensemble-based expected value is a poor statistic. It doesn’t actually predict the value that he would have. There’s no convergence between Bob’s value and “expected value”, so it seems strange to say that Bob “expects to get” the result of the ensemble average here.
You can certainly calculate Bob’s ensemble average, and it will have a higher result than the temporal average (as I state in my post). My claim is that this doesn’t help you, because it’s not representative of Bob’s game at all. In those situations, maximizing temporal average is the best you can do in reality, and the Kelly criterion maximizes that. Trying to maximize ensemble-based expected value here will wipe you out.