On top of what others have said, it’s especially bad in contact sports. A lot of injuries occur when people get hit unexpectedly (can just be a totally unexpected hit, or a hit from a direction/person not expected). With three teams or more, you generally will need three objectives or more (end zones, goals, baskets, whatever). Imagine looking at a football field from the side. Game play largely flows left and right. With a third team, you will either have imbalance or have to add more north south component. This means that game play will have people not coming generally straight toward you, but from angles. Much harder to account for everyone in that scenario.
This is the reason why football has largely outlawed blocks when the blocker is moving in the opposite direction of the runner. It’s near impossible to prepare for and often has devastating consequences.
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