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Imagine this voting algorithm: Each voter ranks their choices. Then, the 1st choice votes are tallied, and the one with the lowest score removed from the running. Those who have the removed contender have their rankings shifted up one, i.e. their second choice is now considered their first choice. Everybody else keeps their rankings.

How does Arrow's Theorem invalidate this? It seems to satisfy all four conditions, but the theorem shows it can't. Which condition does this not meet, and why?

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The voting system you describe is called Single transferrable vote (STV).

Arrow's theorem does not discuss STV specifically. Instead, it states that no electoral system exist that satisfies a certain number of criteria at once.

One of these criteria is monotonicity, and there are studies that conclude STV is non-monotonic.

In two words, about monotonicity.

Fix a certain voting profile (a set of 3+ Candidates, 2+ Voters, and each Voter has already voted).
Now, imagine that during the voting, Alice the Voter gave CandidateX a certain Rank n (in other words, CandidateX is Alice's nth choice).

Now, if the situation is possible that when Alice lowers the X's Rank, and after the votes recount, X gets better outcome from elections, then the voting system is non-monotonic.
The same is for the opposite situation: Alice raises their Rank for CandidateX, and this results for worse voting outcome for X.

There are several articles available that describe this phenomenon:

This opens the way to various techniques of tactical voting; in some corner cases, not voting for your preferred candidate improves their result.

Also, Monotonicity is not the only criterion that STV apparently fails to satisfy.

Good news is that Arrow's theorem does not "invalidate" STV because it states that there is no "perfect" voting system. Or, it does "invalidate" all voting systems, if you will.

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  • Interesting. Would you be willing to expand on how this system could be non-momotonic? I can't understand how Alice lowering the rank could ever make the outcome better. – Duncan X Simpson Jun 5 '17 at 0:25
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    There's a sense in which Arrow's theorem proves that the ideal voting system is one where the strategy is too complicated to easily explain to someone. – eyeballfrog Jun 5 '17 at 0:51
  • @DuncanXSimpson, see the two linked articles, they contain examples with specific numbers how to craft such cases (the "Irish" example is most evident). – bytebuster Jun 5 '17 at 0:52
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    @DuncanXSimpson Intuitively, because it impacts who gets eliminated at each iteration and to who votes "transfer", so you can strategically order candidates to increase the odds that one or more of your preferred choices inherits enough votes. If you like B and C, but they're not the most popular, you might know that B votes tend to prefer C next, and C voters tend to prefer not-B next. So you intentionally vote B low so they get eliminated and transfer votes to C, giving you one of your preferences when you'd have none if C got eliminated. – zibadawa timmy Jun 5 '17 at 1:57
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    Check out Nicky Case's awesome interactive explanation of voting systems, which includes an interactive example of non-monotonicity in IRV (Instant Runoff Voting - STV when there's only one winner) – DMGregory Jun 5 '17 at 2:12
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If only a single candidate can win, this is called Instant Runoff Voting or the Alternative Vote (IRV/AV). If multiple candidates can win, then it is called the Single Transferable Vote.

IRV violates the Condorcet criterion. If there is a Condorcet winner (not always true), then the Condorcet winner wins the race. Arrow's Impossibility Theorem calls this Pareto Efficiency and describes it as "If every voter prefers alternative X over alternative Y, then the group prefers X over Y." I.e. Y can't win the election over X if the majority of the voters prefer X to Y.

In the example that Wikipedia uses, Nashville is the Condorcet winner, preferred 58-42 over Memphis and 68-32 over Knoxville and Chattanooga. But Knoxville wins because Chattanooga has the fewest first round votes, and its first round votes go to Knoxville in the second round. Nashville is eliminated in the second round, and its first place votes go to Knoxville. Knoxville beats Memphis to win.

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