In this case, I'm sorry to say, they mean quite a bit. Whoever loses the "little battles", Arcana-wise, is no longer in play and no longer have the chance available to influence the multiverse as a whole by winning. Those who win the "little battles" can continue on to the next match.

So while I'm sure what you know about war and time and all that may not apply here. It could very well trip you up if you try to approach this situation by the rules you're used to following.