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Once closed down and locked up by Zeus due to the senseless violence, it was reopened when Sora opened the lock during his first visit to Olympus Coliseum. Hades knew Sora would 'play hero' and save Megara so that the Underdrome would be reopened. Hades then decided to use the Underdrome to pit Auron and Hercules against each other much like he wished to do with Cloud in Kingdom Hearts.
The different tournaments you can enter are:
There are also the Paradox Cups, organized by Hades himself. These fights are much more difficult than the normal tournaments. But, the background of the battlefield is green, so, it may be possible that all the Paradox Cups take place inside the River Styx.