Let the fate decide...

Red soon realize that it is his change to be a female without revealing his secret. And he want's the guys competing for him. But what he also know is that they're for sure competing for his mother, who is a REAL women. It'll be embarrassing if they just try every means to lose...

So, why not make it simpler, "Rock-paper-scissors, the winner gets me", says Red.

"What? It's not even a competition." complains the woodsman.

"Of course it is, and the real gamer knows how to win", and how to loose.

It's not what they want to compete. They want to show their strength or something, not luck. But they still accepted your proposal.