  We study contests where the designer's objective is an extension of the
widely studied objective of maximizing the total output: The designer gets zero
marginal utility from a player's output if the output of the player is very low
or very high. We model this using two objective functions: binary threshold,
where a player's contribution to the designer's utility is 1 if her output is
above a certain threshold, and 0 otherwise; and linear threshold, where a
player's contribution is linear if her output is between a lower and an upper
threshold, and becomes constant below the lower and above the upper threshold.
For both of these objectives, we study (1) rank-order allocation contests that
use only the ranking of the players to assign prizes and (2) general contests
that may use the numerical values of the players' outputs to assign prizes. We
characterize the contests that maximize the designer's objective and indicate
techniques to efficiently compute them. We also prove that for the linear
threshold objective, a contest that distributes the prize equally among a fixed
number of top-ranked players offers a factor-2 approximation to the optimal
rank-order allocation contest.
