"Each player ignores a result below their own score and uses that score number instead."
With your explanation in-hand, this line in the rules now makes more sense, but on first read it wasn't clear to me. It might be helpful to use language closer to what you wrote in the comment above.
This is a subtly brilliant mechanic by setting a floor for the character based on their natural ability in that approach, but still allowing the opponent to get lucky and roll higher.