Arrow's impossibility theorem

This is correct, with one addition:

> Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above in the general case.

Nobel Laureate Amartya Sen[0] has demonstrated that, while there is no system that satisfies all four characteristics in the general case, there are systems that either satisfy all four conditions either probabilistically or satisfy all four conditions subject to some very weak assumptions.

The example I've heard him use is of the 2000 election in Florida, with Bush, Gore, and Nader (let's ignore Buchanan for simplicity). While technically there are 3! = 6 possible ways to rank the candidates, in practice, the ranking (Nader, Bush, Gore) is much less likely than (Nader, Gore, Bush) or (Gore, Nader, Bush). If we introduce one minor assumption about the relative frequencies of the rankings, we can prove that instant-runoff voting[1] does always satisfy all four of Arrow's criteria[2].

To use an analogy from computer science, the halting problem is undecidable in the general case, but that doesn't prevent static analysis tools from spotting many infinite loops; it just means it can't spot all infinite loops with 100% accuracy.

[0] https://en.wikipedia.org/wiki/Amartya_Sen

[1] https://en.wikipedia.org/wiki/Instant-runoff_voting

[2] A different example: instead of making assumptions about the relative frequencies, we could make assumptions about the number of axes that candidates may have and the way they cluster around them. This realistically depicts both two-party and multiparty elections in most parts of the world, since political positions are not uniformly distributed along n dimensions.

This is how I was taught it (or understood I was taught it) - at law school, so it might have been dumbed down.

The impossibility is the impossibility of ensuring rational (transitive) outcomes amongst ranked preferences and adhering to a set of fair and democratic norms.

A rational transitive outcomes is one in which votes result in option A being preferred over option B and option B being preferred over option C, such that A is preferred over C (eg, A > B > C). Option A is known as the Condorcet winner.

But there may be cases where the vote yields no Condorcet winner (eg, A > B > C > A). This is illustrated by the following table:

Preferences Voter 1 Choc Vanil Strwb Voter 2 Vanil Strwb Choc Voter 3 Strwb Choc Vanil

Two voters prefer C over V and two prefer V over S, but two also prefer S over C.

To ensure transitivity, we can introduce voting rules, but it is impossible to introduce rules that do not violate the fair and democratic norms (referred to as the pre-specified criteria in the Wikipedia article: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives).

There is a simple explanation in the introduction:

  In short, the theorem states that no rank-order voting system can be 
  designed that always satisfies these three "fairness" criteria:

    1. If every voter prefers alternative X over alternative Y, then the 
      group prefers X over Y.
    2. 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).
    3. There is no "dictator": no single voter possesses the power to 
      always determine the group's preference.

A "rank-order voting system" is one in which all voters order the possible choices, from most preferred to least preferred. The idea is that based on everyone's votes, and some set of rules on how to evaluate those votes, you can come up with a decision for the entire group that respects those preferences, in some way that is considered fair.

All of these requirements listed above seem like fairly simple requirements for a voting system to have; if there's unanimous agreement about the ordering of two of the choices, then the voting system should order them that way. If a voter changes preferences about one choice C (maybe changing between them being ranked above or below D, or above or below one of A or B), without changing their ordering between choices A and B, that shouldn't affect the outcome of the A/B ordering that the voting system gives you. And finally, there is not dictator; no single voter is able to always determine the group ordering unilaterally.

The first requirement seems pretty obvious; if everyone is unanimous that we should pick one option over another, then the group should pick that option.

The second is a little trickier; but it basically states that there shouldn't be able to be "spoiler"; someone whose ordering in people's preferences affects the ordering of to other choices. Think about, say, the 2000 election, with Bush and Gore and a couple of third party candidates; now, that used just single choices per voter rather than a ranking, but imagine it used a ranking. So let's say Bush and Gore are fairly close, but 51% of voters prefer Gore over Bush. Their preferences for other candidates should not affect the fact that Gore wins over Bush. If I rank candidates Nader > Gore > Bush, that should not change the Gore/Bush result compared to if I ranked the candidates Gore > Bush > Nader.

And the last is pretty trivially desirable; generally a democracy wants to democratically make a choice, in which everyone's vote has the same weight in affecting the final outcome.

The fact that these conditions are impossible to achieve together is fairly profound, and means that almost any voting system will have serious flaws under certain circumstances.

As long as politics is not one-dimensional, there are completely reasonable cases where you get a rock-paper-scissors situation between 3 top candidates if voters select whomever is closest to them. That in turn violates the criteria that says the election result can't change if you add a non-winning candiate (rock beats scissors, but when paper enters the race now scissors is computed the winner).

This criteria is called "independence of irrelevant alternatives". A common criticism of Arrow's theorems usefulness is that it is a bit of a stretch to call paper an "irrelevant alternative" when rock-paper-scissors forms a cycle of preferences like that.

If politics is one-dimensional ("single-peaked preferences" in the article), then you never get this situation.

It's also important to note that a much more reasonable criteria exists called "local independence of irrelevant alternatives". This is the idea that total losers joining the race don't affect the result, however someone who is in the top rock-paper-scissors cycle (the "Smith Set") can still affect the result even if they don't win themselves. This is far more reasonable, as it's fairly arbitrary which of the candidates in that top cycle should win.

When there is no top cycle, the Smith Set is a single person (the "Condorcet Winner"). When there is a Smith Set, most reasonable voting systems will pick their winner somewhat arbitrarily as a member of the Smith Set, since everyone in the Smith Set beats everyone outside the Smith Set.

> Granted this doesn't invalidate the Arrow paradox in any way, I'm just saying the paradox isn't true to life because one of its core tenants doesn't quite hold.

However, this plays rather fast and loose with the principles behind Arrow's theorem. First, you're violating the nondictatorial condition (you are the only person voting on your own choice), so it's kind of silly to apply the theorem here in the first place.

Secondly, you're saying that, as soon as Communism becomes an option (but only then), you'd rather die than serve the royal family.

I can see how this might be the case, but if so, you're fundamentally changing the meaning of choice B. It's no longer just the default state ("die of starvation"), but rather dying for a cause (Communism). It's not that the introduction of choice C suddenly means that you're now choosing B over A; it's that it also introduces choice D ("die as a martyr in the name of Communism", as contrasted with "die for no particular cause"). Your ranking is now C > D > A > B.

You have to understand that Arrow's theorem came about as the result of an effort to understand the way that firms make decisions, and the ways that voting structures of boards can influence corporations to make non-optimal decisions for the corporation. Your situation doesn't really apply to the definition of independence of irrelevant alternatives as implied by the context in which Arrow's theorem is applied (or the formal proof thereof).