The Gladiator (Crosstime Traffic #5) by Harry Turtledove
The Soviet Union won the Cold War. The Russians were a little smarter than they were in our own world, and the United States was a little dumber and a lot less resolute. Now, more than a century later, the world's gone Communist, and capitalism is a bad word.
For Gianfranco and his friend Annarita, a couple of teenagers growing up in Milan, life in a heavily regimented, surveillance-rich command economy is just plain dreary. The eventual withering-away of the state doesn't look like it's going to happen anytime soon.
Annarita's a hard-working student and a member of the Young Socialists' League. Gianfranco is a lot less motivated--but on the other hand, his father's a Party apparatchik. The biggest excitement in their lives is a wargame shop called The Gladiator, which runs tournaments, and stocks marvelous complex games you can't find anywhere else.
Then, abruptly, the shop is shut down. Someone's figured out that The Gladiator's games are teaching counterrevolutionary capitalist principles. The Security Police are searching high and low for the shop's proprietors, who've not only vanished into thin air, but have left behind sets of fingerprints that aren't in the records of any government on earth.
Only one staffer is left: Gianfranco and Annarita's friend Eduardo. He's on the run, and he comes to them in secret with an astonishing story: he's a time trader from our own timeline, accidentally left behind when the store was evacuated. The only way Eduardo can get home to his own timeline is if Gianfranco and Annarita can help him reach one of the other time trader sites in this world--and the Security Police will be on their tails all the way there.