If you offered money to everyone if zero won,

That would be an interesting experiment as well, but I was instead suggesting that most entries would be substantially lower (though perhaps still mostly nonzero) if there were a significant monetary prize for the winner (to be split N ways for an N-way tie), and that this new distribution of responses might look like the distribution that would occur without a prize but if it were known that a certain number of high responses would be thrown out.