Mainly because of teaming up.
Consider three teams on the field, each team trying to score. If team A is currently in the lead, team B and C would cooperate more to prevent team A from scoring – even possibly leading to team B helping team C. It could be especially problematic if one of the teams had no chance of winning on their own, but by making deals with one of the other teams, arranges to come in second by playing “kingmaker” – Team A (weak) helps keep Team C from doing anything while Team B scores, and then Team B allows Team A to score some points; all while Team C has no chance of getting anything done because they are outnumbered the entire game.
This is not just a problem with sports, but any game. Board games like Monopoly, Settlers of Catan, Risk, and especially Diplomacy are all at some level based on players forming alliances and teaming up. For this reason, there is an open question among board game players how much of these games are skill, and how much they can be just shortcut with pre-arranged alliances.
Have you ever watched a three way match in pro wrestling? You get people teaming up against one guy, you get someone interrupting every opportunity to score (pin or submit someone). You get refs that are easily distracted by all the chaos. You get a lot of outcomes where one guy(or gal) wins because they just got lucky at the right moment. All this scripted wrestling stuff would happen in a real sport as well.
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.
Most high interactive competitive games are played with 2 sides, not just contact sports. Competitive games with 3+ sides and high interactions are very rare. A big issue with 3+ sides are metagame, where factors outside the game strongly influence the outcome. There are also many harder-to-check ways of breaking rules, like undue collusion and wintrading. Then there are also issue like lame duck (side that already lose but still have to try hard to keep the game balanced) and kingmaker (a side that can’t win but can help someone else win).
Your question has already been sufficiently answered, but if you’re curious about multi-team sports look up omega ball. It’s very strange. Best way I can describe it is like soccer with three teams on a circular pitch. For the curious, you can even watch [full matches on YouTube](https://youtu.be/B3Uh7UTlYak).
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