If you give them the chance, Northwestern PhD students will take a perfectly good game and turn it into a mad science experiment.  First there was auction scrabble, now from the mind of Scott Ogawa we have the pari-mutuel NCAA bracket pool.

Here’s how it worked.  Every game in the bracket was worth 1000 points. Those 1000 points will be shared among all of the participants who picked the winner of that game.  These scores are added up for the entire bracket to determine the final standings.  The winner is the person with the most points and he takes all the money wagered.

Intrigued, I entered the pool and submitted a bracket which picked every single underdog in every single game.  Just to make a point.

Here’s the point.  No matter how you score your NCAA pool you are going to create a game with the following property:  assuming symmetric information and a large enough market, in equilibrium every possible bet will give exactly the same expected payoff.  In other words an absurd bet like all underdogs will win is going to do just as well as any other, less absurd bet.

This is easy to see in simple example, like a horse race where pari-mutuel betting is most commonly used.  Suppose A wins with twice the probability that B wins. This will attract bets on A until the number of bettors sharing in the purse when A wins is so large that B begins to be an attractive bet. In equilibrium there will be twice as much money in total bet on A as on B, equalizing the expected payoff from the two bets. One thing to keep in mind here is that the market must be large enough for these odds to equilibrate. (Without enough bettors the payoff on A may not be driven low enough to make B a viable bet.)

It’s a little more complicated though with a full 64 team tournament bracket. Because while each individual matchup has a pari-mutuel aspect, there is one key difference.  If you want to have a horse in the second-round race, you need to pick a winner in the first round.  So your incentive to pick a team in the first round must also take this into account.  And indeed, the bet share in a first round game will not exactly offset the odds of winning as it would in a standalone horse race.

On top of that, you aren’t necessarily trying to maximize the expected number points.  You just want to have the most points, and that’s a completely different incentive.  Nevertheless the overall game has the equilibrium property mentioned above.

(Now keep in mind the assumptions of symmetric information and a large market.  These are both likely to be violated in your office pool.  But in Scott’s particular version of the game this only works in favor of betting longshots. First of all the people who enter basketball pools generally believe they have better information than they actually have so favorites are likely to be over-subscribed. Second, the scoring system heavily favors being the only one to pick the winner of a match which is possible in a small market. )

In fact, my bracket, 100% underdogs, Lehigh going all the way, finished just below the median in the pool.  (Admittedly the market wasn’t nearly large enough for me to have been able to count on this.  I benefited from an upset-laden first round.)

Proving that equilibrium of an NCAA bracket pool has this equilibrium property is a great prelim question.