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A Note on Collective Decision-Making

This morning, part of a radio news program here in Washington featured a discussion of general elections and runoff elections.  The specific context was Democrat Blanche Lincoln’s prospects for re-nomination by her party to stand in November as the Democrats’ candidate for reelection to the U.S. Senate from Arkansas.  If Ms. Lincoln doesn’t get a majority of votes in today’s three-person race for the nomination – and prospects are poor that she’ll clear the 50-percent-vote hurdle – there will be a runoff election for the nomination: the top two vote getters will then go against each other in this follow-up election.

In light of this prospect of a runoff election, one of the radio commentors said off-handedly something like “In a runoff election, voters’ least-favored candidates are eliminated, leaving only the most-favored candidates to challenge each other.”  (I forget the exact wording, but I’m sure that I capture here the essense of what this guy said and meant.)

This point about the nature of a runoff election is not necessarily true.

Suppose there are nine voters.

Voters 1, 2, 3, and 4 each prefer candidate A to candidate C and candidate C to candidate B.  That is, each of these four voters ranks the three candidates as such: A>C>B.

Voters 5, 6, and 7 rank the candidates like this: B>C>A.

Voters 8 and 9 rank the candidates like this: C>B>A.

In the general elecation, A will receive 44.4 percent of the vote (4 of 9 votes cast); B will receive 33.3 percent of the vote (3 of 9 votes cast); and C will receive 22.2 of the vote (2 of the 9 votes cast).

Because no candidate won a majority of the vote in the general election, a runoff election is held between the top two vote-getters from the general election: candidates A and B.

In the runoff election, candidate B will win 55.6 percent of the vote (5 of 9 votes cast).  B will then be sworn into office, presumably as the voters’ preferred candidate.

But look more closely.  Suppose that the candidate who received the fewest votes in the general election – candidate C – were to run against candidate B in a runoff election.  Which of these two candidates would win?  Answer: C.  In such an election, C would win 66.7 percent of the vote (6 of 9 votes cast), thus trouncing candidate B!  Also note that if a runoff election were to pit C against A – the candidate who received the most votes in the general election – C would also defeat A: C would get 55.6 percent of the vote (5 of 9 votes cast) to A’s 44.4 percent (4 of 9 votes cast).

So is it correct to say that candidate B is the voters’ most-preferred candidate?  Clearly not.


The point of this exercise is to make clear that describing the winner of any fair and honest election as being the ‘choice of the voters’ is fraught with potential inaccuracies.

As Kenneth Arrow showed, every method of making collective choices introduces some arbitrariness into the outcome.  Such arbitrariness is unavoidable; it is not unique to the method of general-election/runoff-election.

To the extent that choices must be made collectively, the unavoidable arbitrariness of collective-choice outcomes is simply a feature of reality that must be dealt with. But let’s not romanticize the outcomes of collective decision-making – including that of majoritarian elections.  These outcomes – no matter how above-board and fair and participatory are the processes that generate them – cannot be described unambiguously as reflecting ‘the will of The People’ or even ‘the will of the voters.’