Or at least that's "a" question -- one that comes up yearly in the Kaggle competition. Here's a version of it that popped up this year.
The Kaggle competition (for those who aren't aware) uses log-loss scoring. Competitors predict which team will win as a confidence level (e.g., 95% certain of a win by Kentucky) and then are rewarded/punished accordingly. And since the scoring is logarithmic, you are punished a lot if you make a very confident wrong decision.
The question that plagues competitors is whether forcing their predictions to be more conservative or less conservative will improve their chances of winning the contest. (Or at least finishing in the top five and receiving a cash prize.) Note that this is only concerned with winning the contest, not with improving the predictions. Presumably your predictions are already as accurate as you can make them, and artificially changing them would make them worse -- in the long run. But the Kaggle contest isn't concerned with the long run -- it's only concerned with how you perform during this particular March Madness.
As a thought experiment, let's assume that you could change your entry right before the final game. You can see the current standings, but not any of the other entries. Would you change your entry? And if so, how?
Well, if you see that you're in first place with a big lead, you might not change it at all. Or maybe you'd make your pick more conservative so that you could be sure you wouldn't lose much if your pick was wrong. But if you didn't have a big lead (and in general the farther away from first place you were) you'd probably want to gamble on getting that last game correct. At that point "average" performance cannot be expected to move you ahead of the team's ahead of you, and even "good" performance might be passed by someone behind you who was willing to gamble more than you.
Since it's much more likely that you will be losing the contest going into the final game than in first place with a big lead, I think this argues that (if your goal is to maximize your expected profit) you should "gamble" on at least the last game. It's left to the reader to apply this reasoning recursively to games before the final game :-).
As a concrete example of this, last year Juho Kokkala submitted entries based upon "Steal This Entry" but with Kentucky's probabilities turned up to 1.0. The non-gambling "Steal This Entry" finished in 42nd place, but if Kentucky had won out, Juho would have probably placed in the top two and collected some prize money.