Is there a statistician in the house?

Suppose I participated in a type of sports "pool" wherein for a small entry fee, I stood a chance of winning a sizable pot of money if I did a better job of picking the winner of, say, a sports tournament of some kind. What is the optimal strategy for picking winners in this tournament, just speaking hypothetically?

Here are some characteristics of the "pool":

- It's a six-round, single-elimination tournament, starting with 64 teams and ending with one champion.
- There is a "tournament selection committee" that seeds each of the teams in the tournament. One can assume that the committee makes optimal use of all publicly available information about the relative strength of the teams, and therefore that a team's true probability of winning is in inverse proportion to its seeding (teams seeded #1 have a higher chance of winning than teams seeded #2, etc.).
- The winner of the "pool" is he or she who predicts the most winners among the 63 games played. In some versions, points are weighted according to round (e.g. you get two points for each winner correctly predicted in the second round, four in the third round, etc.)
- Only the top scorer in the pool wins any money. Everyone else goes home empty-pocketed. This is a "winner take all" market.

If your goal is to maximize your expected score, you should pick the higher seeded team to win in each round (again, I'm assuming that I don't have better information about the relative strength of the teams than the selection committee). But if I plan to participate in this pool, say, 100 times over the course of my adult life (there may be two pools every year, a "men's" and a "women's" pool), this strategy may result in my getting the highest possible score on average, but I may never actually win. The winner, it seems, is always, say, a graduate of Winthrop College who picks Winthrop to go all the way despite the odds. For example. So what I think I want is to maximize the chance that my picks will land me at the far right tail of the distribution of scores. What kind of estimator does this for me?

I'm not sure even how to formulate this problem. Do I have to make some assumptions about the strategies followed by my opponents in the pool? For example, let's say there is a "large" number of participants, and their picks follow some kind of normal distribution where the mean pick corresponds to the official seeding and the frequence by which a team is predicted to win in a particular round is inversely proportional to its seeding. To start with, I might want to think about a simpler pool, one where the winner is he/she who does best in just the first round.

That's all. Huge cash prize to whomever gives me the answer to this question by, say, next March.

(Bonus question: prove that in a single elimination tournament with 2^N teams, the total number of games played is always 2^N-1.)