Suppose Alice and Bob both want to determine whether the other has a crush on him/her, but they only wish to share the information if the crush is mutual. Is there a cryptographic protocol that makes this possible without using trusted third parties? It is safe to assume that they will not lie during their "encrypted announcement" to the other.
You could ask Alice and Bob to rate their "affection" for one another using some scale, then apply a protocol for the Socialist Millionaire's Problem (a variant of Yao's Millionaire's Problem) to determine whether the values are equal.
If the scale is finite (say, in the range from 1-10), then a party can learn some information about the other's value by selecting the max or min. Using an open-ended scale would eliminate this problem, but might make "crush equality" a remote possibility.