The world is divided up into seven continents, each ruled by a family or clan of immortals. Altogether, these immortals number 10,000. They cannot be killed by any means, having the ability to regenerate from any damage completely. However, they possess no other abilities, being just as normal as any human. These immortals rule over mortals, which number in the tens of billions. As the members of the families cannot die, wars between nations are conducted very differently in this world. The families use mortals in proxy wars against each other, pitting their mortals against each other in gladiatorial games.
These games are fought against nations for prestige and bragging rights as a way for immortals to compete. The games are conducted each year, where teams of mortals from one nation participate in contests against mortals from other nations, similar to the Olympics. The teams are supported and funded by the ruling clans. These contests are often deadly, and death is very common, resulting in numerous fatalities every year. At certain special events, the games are meant to simulate actual battles, where groups of mortals go to war with others in a designated, controlled environments with the specific intent to kill the other side.
These contests satiate the immortal bloodlust and desire to compete for superiority with each other. However, it is very obvious that the families are using mortals as pawns in their proxy wars for their own entertainment. This would cause discontent among the masses, which greatly outnumber their rulers. While the family members are truly immortal, they are just like any other human physically. It isn't a stretch to say that an enterprising mortal with desires of grandeur can organize a rebellion and take power from the immortals, imprisoning or otherwise incapacitating them. In addition, their mortal subjects may simply tire of being pitted against each other in pointless wars, risking their lives for those who have the benefit of living forever for their own amusement. They could easily rise up against their oppressors and organize states which are ruled by mortals themselves. In our world, kingdoms are often overthrown by one of those things happening, and most only last a few hundred years at best.
As you can see, the immortal empires are very vulnerable, and their rule can easily be done away with. The families need a way to preserve their rulership over the course of millennia, managing to stay in power while being grossly outnumbered. How can the immortals maintain their power base in a way traditional kingdoms cannot?