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Let Teams Choose Their NCAA Bracket Position

Nate Silver has (another) truly insightful post demonstrating the possible perverse advantage of receiving an 11th seed instead of an 8th seed in the NCAA tournament.
He explains:

[An average] team like Arizona would have a considerably better chance — about two-and-a-half times better, in fact — of winning its second round game and advancing to the Round of 16 as a No. 12 seed than as a No. 8 or No. 9 seed. This, of course, is because it has not yet had to face the No. 1 seed.

He also understands why this is a problem:

I think you need to be concerned any time you create an incentive for losing — which is what the current tournament structure does. For example, a team playing in a conference championship that figured to be a No. 9 seed if it won the tournament but a No. 11 seed if it lost might have some incentive to tank.

But when it comes to solutions, he misses an obvious one (which is somewhat inspired by Scorecasting’s suggestion that NFL coaches bid for possession in overtime).  The NCAA could rank the 16 teams in a bracket  from best to worst as it does now — but then allow the teams (starting with the number one seed) to choose which bracket position they would want to occupy.   The first and second ranked teams would almost certainly choose different halves of the bracket.  The 15th and 16th ranked teams would almost certainly still have to play the top ranked teams (because no other bracket positions would be left).  But we would quickly see whether the 8th ranked team would follow the Silver reasoning and opt for what is now the 12th ranked bracket position to increase its chance of surviving to the Sweet Sixteen.
Silver prefers a solution where the NCAA ranks only the top 6 teams, and the rest of the bracket position are determined by lot.  He envisions a tournament selection show “expanded from one hour to two, with the last half involving Vanna White and Dean Smith drawing ping-pong balls out of a tumbler.”  But I’d prefer for the second hour of the show to be having coaches on the clock choosing in succession their bracket position.  Creating an additional strategic decision might be even more fun than drawing ping-pong balls.