I vaguely remember a short story, I believe written decades ago, where the electoral system was so refined that, rather then polling millions of citizens, it allowed to individuate the single person so average that his opinion reflected perfectly the average opinion of the whole population. I am not sure whether he was just asked his preferences or even made President or some other eminent position.
I found the story I was thinking of, where an "average man" was discovered who by a fluke exactly matched all the averages. It's "Null-P", by William Tenn.
The world had been devastated by a nuclear war, and the population was desperate for a bit of normality, so jumped on this paragon of averageness, electing him president and starting a trend (similar to the theme of stories like "Harrison Bergeron", "The Marching Morons", etc.) where "averageness" is celebrated.
In the end,
the descendants of the "average" people are out-competed by evolutionarily advanced dogs who keep humans around simply to throw sticks.
That is "Franchise" by Isaac Asimov. To quote from the linked Wikipedia article:
In the future, the United States has converted to an "electronic democracy" where the computer Multivac selects a single person to answer a number of questions. Multivac will then use the answers and other data to determine what the results of an election would be, avoiding the need for an actual election to be held.