Well, now I know. And it's not pretty.
Based on this commercial — a rare look into the Hermit Kingdom that is McDonaldland — it appears that the McCheese regime went through the motions of an election, sort of the way the old Soviet Union used to do, and with about the same sense of fair play. Mayor McCheese seems to be running unopposed, but even so, the Hamburglar is busily stuffing ballot boxes. And the real power behind the throne, of course, is Ronald McDonald, Father of the Nation, who tells McCheese what to say, counts the ballots and announces the results.
Actually, this bizarre pastiche of banana-republican politics is one of a series of old McDonald's commercials that are all deeply bizarre and well worth viewing (via Slate).