But it is possible to compare two values from different parties without revealing those two values.
I believe I currently have a solution in my head.
EDIT: I plan to use the millionaire solution (link above) to compare pieces. To guard against players cheating the system by always sending a high piece, both players exchange encrypted piece positions with each other at game start. at game end, the decryption keys are exchanged to determine if they had been telling the truth.
It's rough, but it looks doable.