Ludocracy/Athletocracy
Government by sportsmen, with conflicts resolved through sports. It's commonly said that the Olympics, or sports in general, are a substitute for war. Anthropologists have found this opinion globally in many indigenous cultures, and archaeology confirms many traditional sports are of considerable antiquity. However, it's also clear that tribes, peoples, countries, etc. have no reason to accept the result of a sports game when they feel they could succeed in a war for it.
Therefore the goal becomes how to properly set up a government to make this concept work. I imagine it could only thrive within one culture, or very closely related ones. What I envision is that a hegemonic ruler would force his neighbours--and his vassals/internal contenders--into accepting resolution of disputes via the result of a sports match. This of course requires the culture to have that concept, but it isn't uncommon and has happened globally. The benefits of this are greater prosperity from not having destructive wars, which would let this culture react better to truly external threats (i.e. "barbarians").
This culture would still have armies and a military, but they would be indistinguishable from their athletes. All soldiers would be athletes and vice versa. I envision the "best of the best" forming the actual teams competing in sporting events. The foremost general, or even the ruler himself, would be the coach, although he might do double-duty as team captain and actually get on the field (in any case, he's likely to have been an athlete himself). Lesser athletes would be the supporters. They would stay in the stands and cheer on their team while keeping a close eye on the enemy.
If worst comes to worst, then the supporters will protect their athletes and ruler while driving off the enemy, but this would not be a common occurrence. Presumably a sports match in this society would be just as rowdy as certain teams around the world notorious for the actions of their supporters. It would be half ritual, half entertainment, potentially last days and multiple matches, and there would be much alcohol served, gambling, etc. and the resulting fights you'd expect.
If needed, referees would be priests, possibly from a neutral third party state or a religious institution not seen to favour either party. Or perhaps a sovereign with minimal actual power but great cultural/political/religious influence, like the Japanese Emperor, might serve as referee, or perhaps a "league commissioner" of sorts.
I envision this as hard to work in an international relations context, but within a single confederation or country might be feasible. As war becomes more complex with gunpowder weapons and battlefield artillery, this sort of system will likely break down, so it probably would not work past the 17th/18th century or so. At least not without insanely specific circumstance, like one could imagine a society ruled by super-intelligent AI having humans settle their affairs in sports (and possessing the ability to eliminate cheating).
Potential societies--
Ancient Greece - The ancient Olympics were a hugely important religious festival and actually did play an important political role. It seems very possible to take it further and have society be totally dominated by the Olympics (and similar competitions)
Mesoamerica - The "Mesoamerican ballgame" had deep religious significance and like the Olympics, was used for political purposes. It quite literally did settle disputes between prominent figures and states in certain times and places. However, it was more entertainment than politics in the Aztec Empire, so Mesoamerica would still need some alternate PODs to get the ballgame as the centerpiece of society.
Ancient China - The sport of cuju was first known in the Warring States era. Maybe it gets more popular earlier and the conflicts of that period are resolved over cuju matches?
Medieval Japan - Kemari descends from cuju (and is written with the same kanji) and was popular in the Imperial Court during the late Heian period. It continued rising in popularity in the subsequent Kamakura period and reached the warrior class as a whole in the Muromachi period. One could imagine with the right series of PODs the Shogun and Emperor telling the parties of the OTL Onin War to settle it over kemari instead of repeatedly burning down Kyoto.
Future Earth - We get taken over by AI, but the AI doesn't intervene much in our affairs beside telling us to stop fighting wars and settle it through sports instead
