Voting: a game theory look

[Poofler]

Why do they think no one vote matters?

If it's because your one vote won't decide an election, you esentially are describing the current state of the world where we vote out of civic duty. And without civic duty, the cheating incentive would run up the numbers past just Bob and his neighbor.

If it's because they do not recognize the legitamacy of the process, that specific reason why it is illegitimate matters. If the culprit is rampant corruption of results, invalidating the votes, there really is no incentive, and I don't see how anyone votes. The action would be meaningless. If all voters believe no candidate was different from another candidate, then I see no incentive to vote. But they would still abide by however the existing government decided the election, as they abstained out of indifference and utility, not out of disgust for government.