You can't win with these people. You could convince a few, maybe, but most of them had made their minds up about politics many years ago. It's an almost fatalistic attitude of "things suck, nothing will change, nothing can be done, and I won't even try, so don't ask". These people are the toughest sort to persuade when it comes to voting, but we still have to try I guess.
Why not voting is the thing to do?
Jose Saramago's novel Seeing tells a story of strange events that took place in an unidentified capital city of an unidentified democratic country.
The elections are over and the turnout has been really high. But when the results are announced the government is devastated to discover that over 70% of the votes cast are blank. Not spoiled, not abstained, just blank. They hastily call a new election but the results only get worse, now over 83% have cast blank votes. The Government panics, indignantly struggling to contain what they see as a strike at the very heart of democracy. But there is no sign of where this conspiracy has come from, no sign of what criminal mastermind is behind it all.
500 citizens are seized at random and their families informed that they shouldn't worry about the lack of info on whereabouts of their loved ones: "In that very silence lays the key that could guarantee their personal safety". This doesn't work and the government declares a state of emergency and blockades the City and finally manufactures its own terrorist ring leader. But life continues almost normally as people fight back in a Ghandian non-violent way.
Why is not voting more problematic for any government than voting against it?
Because any government has only so much power that its subject - we - are willing to give it.
"The voter’s abstention goes further than the intra-political negation, the vote of no confidence: it rejects the very frame of decision.
The threat today is not passivity, but pseudo activity, the urge to be active, to participate, to mask the nothingness of what is going on. People intervene all the time, "do something”; academics participate in meaningless debates, and so on.
The truly difficult thing is to step back, to withdraw. those in power often prefer even a `critical` participation , a dialogue, to silence - just to engage us in dialogue, to make sure our ominous passivity is broken. The voter’s abstention is thus a true political act: it forcefully confronts us with the vacuity of today's democracies."
-Slavoj Zizek: Violence