Umm... okay. whatever.
Let's see. "Utility" refers to the measure of satisfaction one gets from a product or service. From that perspective, I suppose you're right; Aggie is going to be happy if he feels like a winner. Quitting when he's ahead makes him feel thusly, so he should do so to increase his utility.
Sadly, utility has little to do with probability and statistics. Whether Aggie is playing his first shoe of the night, or the last one (prior to quitting because he's doubled his buy-in) doesn't make a damn bit of difference to the cards and/or to the odds of him winning. Quitting when he's ahead does him absolutely zero good in this case- his odds aren't increased, nor is his expectation.
This latter thing is, of course, what he was asking about. He didn't ask if he'd FEEL good (utility) if he quit when he was ahead; he asked if he SHOULD quit when he's ahead, if that would give him any advantage in the long run.
Which, put simply, it doesn't.
Call me brainwashed if you like, but your own example merely demonstrates that what I've been saying is correct. If, in your hypothetical game, someone were to hit the big prize... and the house doesn't change the rules... then would it help them to quit as soon as they hit the jackpot?
No. Their odds of winning the jackpot (and thereby increasing their bank) doesn't change one bit once they've hit it. The odds of winning are the same on the very next spin- and indeed, in your example, given enough spins sooner or later that wheel will hit the jackpot *twice in a row*.
When that case happens, the guy who kept spinning after he won (instead of quitting, as in your example) did FAR better than the guy who quit after his first hit.
But BJ is not a hypothetical wheel. Your imaginary game has huge variance; several thousand losses for each win (but boy, what a win!) is an extreme and silly example when you're comparing it to BJ, where the house advantage with perfect basic strategy is still very very small (and the player advantage with a solid count system is also very very small).
Opportunity cost doesn't really play here, either. Opportunity cost, as I understand it (and it sounds as though you're willing to correct me if I'm wrong) is the cost of something lost because of an opportunity that you passed up.
Well, by quitting when you're ahead, the opportunity cost you're passing up is the next shoe. Forking, you've blown off the chance that this cost might be huge.
You don't know what the cost is unless you sit around and watch it; maybe there's a ton of 15s and 16s to the dealer, and you screwed the pooch by quitting early- your oppotunity cost was gigantic. Or maybe the dealer hits his 12 every time with a 9, making 21 and continually cleaning the players' clocks. In that case, you did great by leaving early, because your opportunity cost was negative- sticking around would have actually cost you.
However, advantage play isn't based on a short-term look and opportunity. It's based on the long run, and on knowing that if you play perfect BS and can spread your bets properly according to your counting system, the ODDS of winding up a winner are in your favor.
And those odds are the same for *every single new shoe*. Going in, your odds are always the same; it's the events that happen over the course of the shoe that lead to the variance in how much you wind up winning.
So to wind up, forking, when you say:
[i]"Then everyone asks me: what about tomorrow's or the next session? To which I reply, tomorrow's actual results or the next session's actul results is not known, and if the game opportunity hasn't changed, then the expectation hasn't changed. By quitting today, the player derived some utility and can risk his larger bankroll for the next session, whenever that may be."[/i]
...you're only partially right. Sure, Aggie has attained some utility; he's pleased by his win. Of course, he might have passed up even MORE utility, because that next shoe might have had a bunch of double-down opportunities. Since he's an advantage player, the odds say that over the long run he IS missing out on utility.
(Not to mention that if you're trying to be an advantage player for short-term utility, you're in it for the wrong reason and are setting yourself up for trouble.)
But where you're wrong is in pooh-poohing the expectation. The expectation for an advantage player doing it right is ALWAYS positive. This is what Aggie asked. His original question is whether or not it does him any good- ie, lead to higher winnings- to quit while he's ahead.
The answer to that is no. His expectation is the same going into every new shoe, whether he just cleaned out the rack and they're bringing more chips, or he just got his clock cleaned.
As I said earlier, you're arguing about *feelings*. Utility is a feeling; it's essentially unquantifiable since it's the satisfaction one gets from winning. It's unquantifiable because we have no solid way of knowing for sure if the guy would have won during the next shoe or not.
If you wanna feel good, sure, quit when you're ahead. You'll FEEL great. You might be leaving money on the table, but who cares- you're in it to feel neato.
If you wanna utilize advantage play and work the odds that are in your favor, you have to keep playing; it works over the long run, which means that the more shoes you sit through, the more likely you'll wind up a winner.
Coop