In Pompeii there are poltical slogans written on the walls. Does that mean democracy continued outside the capital even after the fall of the republic? If "yes" how long did it last?

In the city of Rome itself, Augustus and his successors didn't abolish the Senate and the elected offices like the consulship, but gradually reduced their power, ensured that their cronies or relatives were elected to all important offices, etc.

In provincial cities like Pompeii they didn't really change things. The elections for the local aediles and duoviri for the city of Pompeii were highly contested and involved lots of graffiti and negative campaigning. https://campaignstops.blogs.nytimes.com/2012/08/30/the-attack-ad-pompeii-style/

Even today, lots of dictatorships like to run local elections and devolve power to the local level. It's a useful way for people to discuss local grievances, allow local rich guys to satisfy their ambition and it's a more efficient way of running things than referring every piddling little decision back to the capital. There are local elections in China for instance, which are highly competitive - just as long as no one questions the unchallenged beneficent rule of the Communist party. https://en.wikipedia.org/wiki/Elections_in_China

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