Great responses to some of the objections! I can add that we do indeed let the kids perceive it as bidding for putting them to bed ("Mommy wins!" etc). On the other hand, we're usually quite honest with them about our preferences, like "when you're uncooperative at bedtime like this it makes us not enjoy putting you kids to bed".
You're quite right that it's better for the kids to see us bidding to not have to put them to bed than to hear us arguing about it.
(You're also quite right that the primary value of involving money is simply to quantify preferences to determine what's socially efficient. Following through with the payments is needed to keep us honest in expressing our preferences, and also adds fairness by compensating the loser for not getting their way. But that part's more icing.)
You're quite right that it's better for the kids to see us bidding to not have to put them to bed than to hear us arguing about it.
(You're also quite right that the primary value of involving money is simply to quantify preferences to determine what's socially efficient. Following through with the payments is needed to keep us honest in expressing our preferences, and also adds fairness by compensating the loser for not getting their way. But that part's more icing.)