There are a lot of games about winning elections. There are a lot of games about international diplomacy and wars. As far as I know, there is only one Republic of Rome.
You play a faction of senators in the Roman Senate (264-43 AD). You play for your own faction, but if Rome suffers a disaster than you all lose. Wars, financial mismanagement, and failing to keep the population content are all ways for everyone to lose.
The focus is on negotiation and politics of the senate. Wars are fairly abstracted, but the senator who leads troops into battle (making them veterans) can always return home to try and conquer Rome by force of arms (as a counter-weight senators out fighting can't vote on important matters of state while they're gone).
Political assassinations, gaining governorships (to corruptly funnel money to yourself), and other forms of corrupt government can all be prosecuted by the Censor (but whoever controls the senator elected to this position controls who gets prosecuted and who goes free).
You can call for land reform (basically a tax cut which continually cuts into the budget). They are very popular with the people and raise the politician who calls for them. Anyone who votes against it loses popularity. Forcing players who are leading in popularity to choose between their lead and Rome's interests is priceless.
Valley games is reprinting this classic game later this year. It's another game on my "can't hardly wait" list.