There are some mathematical theorems regarding the principles of voting systems that are well-known in popular science. Take, for example, Arrow’s impossibility theorem, which states that no perfect voting system exists when choosing among three or more options, or the closely related Gibbard–Satterthwaite theorem1. There is, however, a basic claim that is not so popular, although quite interesting in my humble opinion: May’s theorem. It states that there is no other reasonable voting system for two candidates than a simple majority rule.

Preliminaries and a formulation of May’s theorem#

A reasonable voting system for two candidates should satisfy exactly four criteria:

  • Near decisiveness — it should choose exactly one winner in every situation except when both candidates receive the same number of votes.
  • Anonymity — all voters should be treated equally.
  • Neutrality — all candidates should be treated equally.
  • Monotonicity — if candidate $A$ wins, transferring votes from candidate $B$ to candidate $A$ should not cause $A$ to lose.

Meeting three of these criteria is relatively easy; for example, a dictatorship (where the result depends on only one voter) is decisive, neutral, and monotone. We are, however, interested in systems that meet all four criteria at once.

May’s theorem says that the only voting system that is nearly decisive, anonymous, neutral, and monotone is majority rule (where the candidate with over half the votes wins).

Proof#

Our social choice function (voting system) should be anonymous, meaning that the identities of individual voters are irrelevant—we care only about the total number of votes each candidate receives. Let $a$ and $b$ be the numbers of votes for candidates $A$ and $B$, respectively.

In the special case where $a = b$, every neutral voting system should declare a tie.

If $a \neq b$, we can assume without loss of generality that $a > b$, and aim to show that candidate $A$ must win. Since the voting system is nearly decisive, it suffices to prove that $B$ cannot be the winner.

We proceed by contradiction — suppose $B$ is the winner. Since $a > b$, we can increase the number of votes for candidate $B$ (denoting this number by $b^\prime$) so that $b^\prime = a$, and thus: $$ a^\prime = a - (b^\prime - b) = a - (a - b) = b. $$ By monotonicity, $B$ remains the winner after gaining votes. However, by neutrality, if $B$ wins with $b$ votes, then $A$ should win with $a^\prime = b$ votes — which leads to a contradiction.

Update#

In July 2025, I formalized this theorem in Lean. During the process, I discovered a simpler proof, so I rewrote the argument and updated this blog post. Previously, the proof was split into two cases based on the parity of $a + b$, but it turned out that this distinction was unnecessary.

More to explore#

If you found this topic interesting and are already familiar with Arrow’s impossibility theorem, you might want to check out another one: the Balinski–Young theorem. Unlike Arrow’s, it doesn’t deal with methods of selecting candidates, but rather with methods of apportionment (how many seats a party or a state should receive). It turns out that’s impossible too.

  1. There is a great paper by Philip J. Reny that provides formal proofs of those theorems: Arrow’s theorem and the Gibbard–Satterthwaite theorem: a unified approach. ↩︎