An anomaly concerning ties in lotto-like games

Consider the following game: Each of the n players independently picks a number from 1 through m according to a given probability distribution, and the winner is the player who selected the lowest of the n numbers. A tie occurs if two or more players both selected the lowest number. Surprisingly, contrary to intuition for certain distributions (including one which arises in a version of Bingo, the probability of a tie does not increase montonically. However, as we show the tie probability does increase monotonically when the probability distribution is monotonically non-increasing.