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.
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.