Pick outcome with highest Kelly bet and bet on it consistently (I am not sure if this is the best strategy as opposed to some mixed strategy involving outcomes with different Kelly bets).
seems correct, no mixed strategy needed for games without opposing strategy.
Assign p<1p<1 as the probability that you would continue the game for the next round. If p=1p=1, you would be trapped in the Casino for eternity. If p<1p<1, you would almost surely leave the Casino at some point. This satisfies the requirements of eventually leaving the Casino.
This confuses me—you claim that the player is immortal and fatigue-free, and that he values money linearly with no upper bound. What’s this requirement to leave? If money is NOT valuable in itself, but only in the outside world, you have to add that conversion to your Kelly calculations, including declining marginal utility, which probably means you leave when no bet has a positive Kelly bet size.
Your setup says that money has linear utility, implying that we should ignore any real-world effects like value conditional on leaving. It also says we’re immortal which implies that “option to leave” is as good as actually leaving. You’ll have to change one of these assumptions before you can quantify the motivation to stop playing when there is any +EV bet to be had.
You don’t need memory to evaluate the marginal utility, you just need the static function that converts quantity of money to amount of utility. The memory is entirely in the starting money for the round. At some point, there will be no available bet where your net utility gain is higher by staying than by leaving.
Your first thought:
seems correct, no mixed strategy needed for games without opposing strategy.
This confuses me—you claim that the player is immortal and fatigue-free, and that he values money linearly with no upper bound. What’s this requirement to leave? If money is NOT valuable in itself, but only in the outside world, you have to add that conversion to your Kelly calculations, including declining marginal utility, which probably means you leave when no bet has a positive Kelly bet size.
Money is only valuable in the outside world. So you’ll need to eventually leave the Casino.
You have no memory of previous rounds, so how would you evaluate the declining marginal utility of money?
Your setup says that money has linear utility, implying that we should ignore any real-world effects like value conditional on leaving. It also says we’re immortal which implies that “option to leave” is as good as actually leaving. You’ll have to change one of these assumptions before you can quantify the motivation to stop playing when there is any +EV bet to be had.
You don’t need memory to evaluate the marginal utility, you just need the static function that converts quantity of money to amount of utility. The memory is entirely in the starting money for the round. At some point, there will be no available bet where your net utility gain is higher by staying than by leaving.