### Why can't voting be fair if there are more than two alternatives?

• 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?

Source on this? I don't see how it could possibly favor his or her choice... well, unless you are talking about electoral votes(US)

@Nick122 In a parliamentary system like the Norwegian one can essentially give a negative vote to a party by voting for a party that promises not to cooperate with the given party. I don't see that as a bad thing, though.

Yes, but you will give a negative vote by voting for your candidate as well while even further advancing him.

9 years ago

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).

It sounds like this only applies when voting a single person (e.g. a president), not when voting for a parliament through proportional representation.

If you elect the Condorcet winner, then you eliminate the spoiler effect. Electing the Condorcet winner fulfils all the Arrow criteria; the problem arises when there isn't a Condorcet winner, ie, there's a cycle (Alice beats Bob beats Carol beats Alice). Most real-world elections do have a Condorcet winner.

I'm convinced that Arrow's theorem is wrong because ending up in a cycle is the correct result; the people are irrational and therefore the result is irrational.

Arrow's impossibility theorem only applies to *ranked-order* voting systems. Cardinal voting systems like Range Voting can and do satisfy all of Arrow's criteria: https://governology.wordpress.com/2017/09/05/kenneth-arrow-is-a-dick/

@Joshua - It's a mathematical theorem. It isn't wrong, although it may not be applicable to all aspects of the problem.

@BT Sorry, Gibbard's Theorem does apply to cardinal voting. And cardinal voting simply reduces to approval voting if the voters are rational.

@Acccumulation Yes Gibbard's Theorem does apply to cardinal voting, but it doesn't prove the same properties as Arrow's theorem. Specifically, while Arrow's Theorem proves that ranked-order voting systems cannot have both monotonicity and independence of irrelevant alternatives. Cardinal voting can satisfy both constraints simultaneously. What Gibbard's Theorm proves is that no voting methods are immune to strategic voting. You're also wrong that cardinal voting reduces to approval if voters are rational. So "sorry", do more research.

@BT The original question says "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." Gibbard's Theorem shows that that is true for cardinal voting. And you don't give any justification for your claim that cardinal voting doesn't reduce to approval. That's a rather rude tone for a comment with no support. What "research" am I missing?

@Acccumulation My comment was that Taymon's answer unhelpfully talks about Arrow's Impossibility theorem when it doesn't apply to "any voting system with more than two choice" (as the OP asked). You seem to be arguing against a point I did not make.

@Acccumulation Re strategy, the electorate as a whole does better under score voting when more voters are honest. This isn't true of systems like plurality. This is a clear incentive for rational people to vote honestly and encourage others to vote honestly. https://electionscience.org/library/score-voting-threshold-strategy/ Also, not all cardinal systems provide an incentive for voters to cheat: https://www.accuratedemocracy.com/l_stdscr.htm