On the Strength of May and Arrow’s Theorems

In a literature of impossibilities, intransitivities and  cyclicalities May’s theorem provides a positive result. It says that social decisions between two options, simple majority rule uniquely satisfy four appealing conditions (unrestricted domain, anonimity, neutrality, positive responsiveness).  One interesting way to think of May’s theorem is as the inverse of Arrow with caveats. Arrow’s theorem states that there is no social welfare function that satisfies unrestricted domain, Pareto, IIA and non-dictator (this is the 1963 version). The initial (1951) version had 5 conditions of fairness (unrestricted domain, monotonicity, nonimposition, IIA and non-dictator).

To me it is tempting to ask which result — the `good’ or the `worrying’ — is strongest. For that, one needs to find the conditions that are the weakest or least demanding. I explain below why Arrow’s theorem is stronger than May’s.

  • May’s positive responsiveness (or strong monotonicity) is stronger of a condition than Arrow’s monotonicity. Why? Essentially May’s condition posits that a change in preference favourable to one alternative (call it X) works in favour of X, whereas Arrow’s condition implies that such a change in preference simply does not work against that particular alternative (X).
    •  This means that there may be cases in which Arrow’s monotonicity is satisfied but May’s positive responsiveness is not.
  • May’s anonimity is stronger than Arrow’s non-dictator. To understand why, consider cases where Arrow’s condition is satisfied but anonomity is not. For example,  this happens under weighted majority rule with unequal individual weights.

As such, given that at least 2 conditions are stronger under May’s theorem than under Arrow’s makes the former a weaker theorem than the latter.