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.

(The classic reference is Radner. More recently these ideas are being used in repeated games.)

## 7 comments

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December 22, 2010 at 4:17 am

Thijs#3: unwaveringly *nice*, right?

December 22, 2010 at 9:49 am

jeffThanks for your comment. I see it wasn’t clear so I updated it a bit. And I added a paragraph at the end.

#3 says that *if* you are not going to be nice all year long then the next best strategy is to be naughty all year long. There is no gain from any middle ground strategy.

December 22, 2010 at 11:00 am

ThijsMuch better, thanks. Now I’m all set for next year.

December 23, 2010 at 9:50 am

kwhy does Santa want everyone to be nice as long as possible? It seems this assumption is implicit in the argument. Am I thinking about this correctly?

December 23, 2010 at 10:24 am

jeffThat is the implicit assumption, yes.

December 23, 2010 at 3:06 pm

Lones SmithI thought Santa was *always* watching you, and thus had no monitoring problem. Now I feel hoodwinked by my parents, and wil have to re-evaluate other possible myths I was benighted by. 😦

December 24, 2010 at 11:01 am

kwhat if you could enter into an agreement with Santa to be naughty for a while – because you got to get it out, you know, act like a lunatic on Saturday night – but credibly promise to be nice for longer (or at least, as long as before)…this way the principal (Santa) isn’t an overriding dictator but a friend, sort of.

Of course this (still) carries the implicit assumption that Santa wants you to be nice at least for a while. But, perhaps, this is a better assumption than Santa wanting you to be nice all the time.