Arrows impossibility theorem states:

no rank-order voting system can be designed that satisfies these three "fairness" criteria:

a. If every voter prefers alternative X over alternative Y, then the group prefers X over Y.

b. 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).

c. There is no "dictator": no single voter possesses the power to always determine the group's preference.

Note - there is no time dimension to this theorem. Its modelling what happens on the day of voting in an election.

Given there are many rank-order voting systems (ie democracies) in the world, what theoretical outcomes should we expect given the theorem? It seems to me that the most likely outcome will be breaking the third option, but manifested rather in the existence of an oligarchy - that is a group (small or large) have the power to determine the outcome of the election. This does not mean that they explicitly act together to hijack an election, but simply that their actions are highly correlated through some other means.

Is this actually borne out in experience?

  • If there is no time dimension, then what do you mean by "unchanged"? – Niel de Beaudrap Jul 5 '13 at 11:30
  • if every individual ranks a>b, then the group ranks a>b, so the ranking is unchanged. – Mozibur Ullah Jul 5 '13 at 13:41
  • "If every voter's preference between X and Y remains unchanged [...] even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change" (emphasis mine) – Niel de Beaudrap Jul 5 '13 at 13:48
  • I expect it means that there is no correlation between the variables. That is 'change' should be interpreted as 'different'. – Mozibur Ullah Jul 5 '13 at 13:52
  • I don't think so. From Wikipedia (under Independence of irrelevant alternatives): "The social preference between x and y should depend only on the individual preferences between x and y (Pairwise Independence). More generally, changes in individuals' rankings of irrelevant alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset." Thus: if the preferences within a subset do not change for individuals over time, it also does not change for the consensus. – Niel de Beaudrap Jul 5 '13 at 13:57

(b) happens often enough.

"I prefer Gore to Bush. But, I just heard Nader talk, and now I prefer Nader to Gore."

If you have 51% support for Gore, 49% support for Bush, and 0% support for Nader, but some Gore supporters decide they like Nader even more than Gore, the numbers will go to e.g. 48%, 49%, 3%, and Bush will win the election, even though nobody changed their Gore vs. Bush preference.

(The current system in the U.S. violates (b) a lot more than is mandated by Arrow's theorem--an alternate system wouldn't have this "spoiler" effect manifest quite so easily.)

  • The fairness criteria presumably applies at the time time of balloting, so time doesn't come into it. Of course one could model theoretically a time based voting system to model changes in behaviour over time. – Mozibur Ullah Jun 5 '13 at 4:05
  • 1
    @MoziburUllah - I am using time to illustrate "change". I could rephrase using alternate realities instead. – Rex Kerr Jun 5 '13 at 6:26
  • I accepted your answer - I hadn't understood the point you were making. What do you mean by an 'alternate' system? – Mozibur Ullah Jul 5 '13 at 15:11
  • @MoziburUllah - One could use approval voting or the Condorcet method or Instant Runoff Voting or any number of other systems that have you either rank candidates or divide them into "okay" and "not okay". This way, when Ralph Nader failed to win the Presidency, the vote scoring would recognize that it is still true that more people prefer Gore to Bush. In particular, "strategy-resistant" voting systems are nice because you can just vote for who you want without worrying that you're wasting your vote (or hurting yourself). For much more, see en.wikipedia.org/wiki/Voting_methods – Rex Kerr Jul 5 '13 at 17:51

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