A few thinkers have claimed that your vote effectively influences others like you into behaving in a similar way, magnifying the power of your vote. E.g. see Ben Goertzel's article on the topic of why people bother to vote.
However, this raises more issues - associated with whether there are enough folk like you in relevant ways to swing an election in your favor. While this is a different sum to the one considered by classical game theorists, it doesn't look as though it is going to make it worth voting in a national election.
Why then do so many people vote? The UK recently saw a 72.2 percent turnout in a national referendum about leaving the EU. That's an amazing turnout. The answer seems simple: people are manipulated into voting by politicians and memes. Cultural evolution is fast and powerful and quite capable of manipulating humans into acting against their own best interests. Politicians harness these memes for their own benefit.
Indeed, perhaps democracy is not really about aggregating preferences by voting at all, but rather is a scheme designed to stop peasants from revolting by giving them the illusion that they have a say in how the system is run.
Rather than try and think of ways to make massive national elections make sense, I am inclined to think that another approach would be useful. One proposal for motivating people to vote involves magnifying the importance of each voter - by limiting who can vote. For example, Robin Hanson has proposed this sort of scheme here. However, people seem to think that this solution is somehow not very democratic.
China has another approach which seems better to me: hierarchical elections. Essentially, people elect local town councillors who vote in city selections. City leaders vote in state elections, and state leaders vote in national elections. This way, most individuals vote for a local councillor in a small election which they might plausibly care about enough to bother voting. At each level of the hierarchy the number of voters is relatively small, meaning that each vote has a bigger chance of influencing the result - and so people are more likely to vote and deliberate on their vote.
Hierarchical elections may not solve every election problem - but they seem like a step forwards to me. A country where voting doesn't make much sense is probably not the best sort of democracy to live in. Technology should make voting easier, but we could also be working on structuring elections more intelligently. I think that simulation and experimentation related to this idea would be worthwhile.