This still doesn't make sense to me outside of the specific context.
In the game I ran yesterday, the PCs tried to leave a temple complex they raided, just to find 10 or more armed and armored people waiting for them outside. They slammed the door shut, barred it, and ran away to find another way out. They minimized danger by running away but if they had been less hurt or depleted they may have stayed and fought using the doorway to their defensive advantage and drawn people in a bit at time to try to overwhelm them, etc. If they were higher level or feeling confident the guards were significantly weaker than them, they might have marched right out there and fought OR challenged them to a one-on-one fight between leaders. . .the possibilities are endless.
As for the opponents, they had been ordered to try to capture the PCs, but were give the authority to kill them if necessary, so that also colored how they'd approach the fight.
Sport vs. War just doesn't seem dynamic enough a way to look at the game as I play it at least.