During the Antiquity, many battles were organized in arenas in ancient Rome and ancient Greece. The gladiators fought with wild beasts or slaves.
In arena fighting games, you will always be fighting in an arena, but the environment will not necessarily Roman or Greek. Even the gods can fight in an arena, as in the game Tactical Combat. You choose your divinity, you assign skill points to her and you put her in the arena to fight against another one!