After the rebellion, Panem was overturned. President Paylor didn't want to keep the names given by the Blood Administration, which was what the government that held over the course of Hunger Games had been called. Panem needed a new beginning.
The government advisees drew inspiration from the Blood Admin, of course. It was a central part of history, and shaped the country into what it is today. The experiences of people from those times made sure that the country turned into a happy, peaceful, place, one that did not force two children into an arena every year.
People read history books. Ones that would have been punishable by death if Snow had still been alive. When people learned about the Ancient Roman times, they saw so many similarities. Right from the start. Panem et circenses. Bread and circuses. Well-fed, entertained by the Hunger Games, it was the perfect description of any Capitolite. And Panem was right there in the phrase.
Once everyone had figured that out, they learned more and more. They read about gladiators in the arena. A fight to the death among slaves, while thousands of people cheered. The Gamemakers had followed everything right down to the last little detail. It was called an arena. Underneath the arena a labyrinth ran, where everything came from. The gladiators, rising from the ground. Then the wild animals, released from underneath. People cheering, not bothering to rebel because they're perfectly happy as long as they're watching slaves kill each other. The ones who make it are suppressed by the money and the popularity. Why start a rebellion? Everything about gladiators screams tributes. Everything about it screams Hunger Games.
So they renamed the Capitol. It became Rome, capital of Panem. The Districts each got new names. District One became Rimini. Three is Piacenza. Seven is Forli. Eleven is Salerno. Twelve became Florence. Thirteen, as some inside joke between nerds, was named Pompeii. Most became vibrant cities, but some remained towns. The ones with the smallest populations to begin with were towns. District Eight was almost decimated in the rebellion, so Bologna remained a town. Same with Florence.
All the old arenas were destroyed, except for the one from the Final Hunger Games. Those were the Capitol games. Every Capitolite of reaping age who had not contributed to the winning side of the rebellion was entered. There were no tesserae available, and volunteering was not allowed. To make things "fairer", each child got a certain "ranking"; it was based on what your sibling's, mother's, father's, grandfather's, or grandmother's job was. For example, President Snow's granddaughter was a Level One. All the Gamemakers' children and grandchildren were Level Twos (except for Plutarch), Games announcers and stylists had children marked as Level Three, and so on right down to Level Twelve. These were the lowliest of the low, the children of janitors and bus drivers. The only non-rebel who was spared was Caesar Flickerman, because no one could bring themselves to doom the life of such a likable man's child. All the victors felt that he had helped keep them alive, by making the interview as easy as possible. This, by extension, got them sponsors that probably saved many lives.
Of course, President Snow's granddaughter was reaped. Claudius Templesmith's son. The current District Two stylist's younger brother. Of course, one girl and one boy was reaped who was strong enough and skilled enough to win, but who didn't have any important relatives. Everyone knew that the government would help these kids as much as possible. They wouldn't suffer through mutt attacks. They'd be given their weapons of choice, right in front of their pedestals. Between them would probably be Snow's granddaughter and any other easy, very obviously pro-Snow targets. Panem counted on the fact that one of them would win. They both died in the bloodbath.