It sounds so simple: you’re nice you make the list, you’re naughty you get a stocking full of coal. But just how much of the year do you have to be nice?
It would indeed be simple if Santa could observe perfectly your naughty/nice intentions. Then he could use the grim ledger: you make the list if and only if you are nice all 365 days of the year. But it’s an imperfect world. Even the best intentions go awry. Try as you may to be nice there’s always the chance that you come off looking naughty due to misunderstandings or circumstances beyond your control. Just ask Rod Blagojevich.
And with 365 chances for misunderstanding, the grim ledger makes for a mighty slim list come Christmas Eve. No, in a world of imperfect monitoring, Santa needs a more forgiving test than that. But while it should be forgiving enough to grant entry to the nice, it can’t be so forgiving that it also allows the naughty to pass. And then there’s that dreaded third category of youngster: the game theorist who will try to find just the right mix of naughty and nice to wreak havoc but still make the list. Fortunately for St. Nick, the theory of dynamic moral hazard has it all worked out.
There exists a number T between 0 and 365 (the latter being a “sufficiently large number of periods”) with three key properties
- The probability that a truly nice boy or girl comes out looking nice on at least T days is close to 100%,
- The probability that the unwaveringly naughty gets lucky and comes out looking nice for T days is close to 0%,
- If you are being strategic and you are going to be naughty at least once, then you should go all the way and be unwaveringly naughty.
The formal statement of #3 (which is clearly the crucial property) is the following. You may consider being naughty for Z days and nice for the remaining 365-Z days and if you do your payoff has two parts. First, you get to be naughty for Z days. Second, you have a certain probability of making the list. Property #3 says that the total expected payoff is convex in Z. And with a convex payoff you want to go to extremes, either nice all year long or naughty all year long.
And given #1 and #2, you are better off being nice than naughty. One very important caveat though. It is essential that Santa never let you know how you are doing as the year progresses. Because once you know you’ve achieved your T you are in the clear and you can safely be naughty for the remainder. No wonder he’s so secretive with that list.