I'm Ulysses, and this challenge to fight could cause a problem for the Greek army. I know there are solutions, but I don't know what would be the best decision.
Whoever loses has the ugliest wife!
We could follow rank, and have Achilles, our best fighter, meet the challenge. The problem is, Achilles already thinks he is too good to fight, and if he won, it would only make him more prideful and lazy.
I can fight!
We could also let Ajax fight. It would make our army look good if we lost, because we could say he wasn't our best fighter, but it would go against our ranking.