American idol: should it be a singing contest or a popularity contest?

In the very popular FOX TV reality show, American Idol, the judges, who are presumably experts in evaluating singing effort, have no voting power when the field is narrowed to the top twenty-four contestants. It is only the votes of viewers that count. In the 2007 season of the show, one of the judges, Simon Cowell, threatened to quit the show if a contestant, Sanjaya Malakar, who was clearly a low-ability contestant, won the competition. He was concerned that the show was becoming a popularity contest instead of a singing contest. Is this a problem? Not necessarily. I show that, under certain conditions, making success in the contest dependent on a contestant's popularity and not solely on her singing ability or performance, could paradoxically increase aggregate singing effort. It may be optimal to give the entire voting power to the viewers whose evaluation of singing effort or ability is noisier.