Now then, then now... let me think of a few stand-off contests:

• A good old staring contest - you blink: you lose!
• "Who gives up first" - is a well know game from Niflheim.
  But I am not sure if this works for your game, though. Here you win by giving up first and thenceforth both players lose at all matches.
• A serious cuddling competition - a simple but fateful game - you cuddle to the death, last man standing style.
• A round of Meta-Vagabonds - the competing Vagabonds will have to come up with a few random numbers to describe a Vagabon-Vagabond. That sounds like a creative challenge... but the winner is actually chosen by a random dice roll...