I've heard that mathematically it can be shown that given any voting system with more than two choices, voters can cheat the system by not voting their true opinions in order to game the system and help their first choice.
Why is this the case and what would be an example of its application. How can democracies take this into account? Is it a real threat to fair elections or just a theoretical one?
The mathematical phenomenon you're talking about is Arrow's impossibility theorem. The wiki article has an informal proof.
Specifically, the theorem states that there's no way to design a voting system such that all three of these criteria hold:
- If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
- If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
- There is no "dictator": no single voter possesses the power to always determine the group's preference.
(The theorem is stated in terms of a rank-order voting system. The winner-take-all and first-past-the-post systems are degenerate cases of this; each voter is asked only for their first preference, and the candidate who is the first preference of a majority or plurality of voters wins.)
Every democratic voting system fulfills the first and third criteria; therefore, they have to dispense with the second. This means that (for example) even if 60% of voters favor Candidate Alice over Candidate Bob, it's still possible for Bob to beat Alice. In a winner-take-all or first-past-the-post system, the way this would happen is through the introduction of a third candidate. If 30% of the voters now prefer Carl as their top choice, and all of those voters previously favored Alice over Bob, then Bob now has 40% of the vote while Alice and Carl each have 30%; Bob has beaten Alice. Voting systems such as instant-runoff voting can ameliorate this, but according to Arrow's impossibility theorem, they cannot entirely eliminate the possibility of a situation where Bob beats Alice even though a majority of the voters prefer Alice over Bob.
The most important practical implication of this is that a democratic voting system can't entirely eliminate the spoiler effect. More broadly, it means the possibility always exists that it's in a voter's interest to vote in a way that doesn't reflect their true preferences; in other words, tactical voting is always a factor in elections. Most people in democratic countries accept this as an unfortunate reality of the system. Even so, some systems (such as winner-take-all) are more heavily impacted by tactical voting than others (such as instant-runoff).
You're referring (I think) to Arrow's impossibility theorem, and "voting can't be fair" is an extremely common misinterpretation. What it actually says is that no rank-order voting system can simultaneously satisfy all of Arrow's criteria for the ideal voting system.
First of all, the theorem only applies to rank-order voting, which is what most people are used to but not the only option out there. Rank-order voting is when you sort people in comparison to one another. FPTP is an example of rank-order voting, although the only ranks are "#1" and "everyone else". Another well-known system is the Borda count, where you rank everyone on the ballot, 1 through N.
However, there's a totally separate class of voting called rated voting, where you judge each person individually. For example, on this site you don't sort the posts from best to worst; you take each one and either upvote it, downvote it, or abstain. This is essentially range voting (with the range set between -1 and 1), which is a type of rated voting. Arrow's impossibility theorem says nothing about rated voting, so it's possible for a voting system in this category to be fair, and indeed range voting has many advocates (I won't go so far as to say it's "best" because I'm not sure the experts will ever agree on that).
Besides all that, Arrow's original criteria are quite strict. In particular, independence of irrelevant alternatives (adding a new candidate to an election shouldn't change the result unless they win -- it shouldn't result in another candidate that was in the election suddenly stealing the win from the original winner) is difficult to fully satisfy. Unfortunately, without it the system tends to suffer from the problem you mentioned upfront, strategic voting. For example, the Borda count is a rank-order voting system that satisfies all of Arrow's conditions except IIA, but is almost hilariously vulnerable to strategic voting -- you always want to rank your candidate's strongest opponent last, even if you actually like them second best.
As to the last part of your question, using a weak voting system is definitely a real threat to fair voting, but Arrow's theorem doesn't just boil down to "democracy is impossible". There are many voting systems that are quite resistant to strategic voting, vote splitting, etc. The real problem is that we tend to not use them because they require more work. Plurality voting is attractive in that it requires checking one box, so it ends up in common use despite its problems. In short, we shouldn't worry too much about whether or not a voting system is perfect; picking one that's pretty good would be a vast improvement over the current situation in most elections
I think this is referencing the Gibbard-Satterthwaite theorem, rather than Arrow's Impossibility theorem.
In any electoral system which elects a single winner from amongst three or more candidates, then the system is subject to tactical voting. Tactical voting is where e.g. the system says "please vote for your favourite candidate" but you vote for your second-favourite because doing so makes it more likely for your least favourite to lose.
The formal definition is that if you know how everyone else is going to vote, there are circumstances where you switch your vote away from your sincere preference list and the winner will be someone higher up your preference list than it would be if you voted your sincere preferences.
A good example of this is the Presidential election in Florida in 2000. Many voters voted for Ralph Nader; they would (mostly) have preferred Al Gore to George Bush, but they preferred Nader even more, so they voted for him. However, if they had voted for Gore (in spite of him actually being their second preference) then he would have won, and they would have got their second-best candidate, rather than their worst (well, Pat Buchanan ran in that election, so probably their second-worst in practice).
In real-world elections, there are some systems where is is much harder to vote tactically than others - you never have perfect information on everyone else's vote, so the more information that you need to be able to vote tactically, the less likely people are to do so.
Aside: technically, you can have systems that aren't subject to tactical voting. If you eliminate all but two candidates before the votes are counted (but after they are cast), then you reduce the remaining election to a two-candidate single-winner election. Of course, that means that a candidate that every single person voted for can lose. The actual proof shows that the only exceptions are a dictatorial system (i.e., one voter decides who wins) and a system where some candidates cannot win, even if everyone votes for them.
Let's see this example.
There are 3 voters (1, 2 and 3) and 3 possible decisions (A, B and C).
1st voter thinks that A is better than B and B is better than C
2nd voter thinks that B is better than C and C is better than A
3rd voter thinks that C is better than A and A is better than B
Any voting system which can make any result decision in this situation is unfair. Any fair voting system cannot make any decision in this situation.
It theory, voting system must make correct result decision in any situation. That's why any voting system is unfair.
