# What evidence is there that Arrows Theorem manifests itself in democracies?

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"? Jul 5 '13 at 11:30
• if every individual ranks a>b, then the group ranks a>b, so the ranking is unchanged. 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) 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'. 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. Jul 5 '13 at 13:57