16 January 2019

L. Alan Winters CB, Professor of Economics and Director of the UK Trade Policy Observatory

This note supplements an article on ‘Organising a three-way referendum’ published on The Economist website (16^th January 2019). It offers a worked example to show how the three main approaches to three-way ballots operate and some of the challenges they throw up. It reinforces Ken Arrow’s result that there is no ideal way of combining individual preferences to select one of three options.

I consider three ways of conducting a three-way vote:

(1) A two-round vote, which would ask in:

Round 1: “If  we leave the EU, would you prefer ‘no deal’ or ‘the prime minister’s deal’?”

And then, when the answer is known, to ask (say, a month later to allow time for campaigning):

Round 2: “Now you know how we would leave, would you prefer to leave or to remain?”

This ordering solves the fundamental flaw of the referendum in June 2016—that no-one knew what “leave”’ meant. It is clearly preferable to the alternative ordering which asks first whether you want to leave and then later (or at the same time) which leave option you prefer.

(2) An alternative vote, which has all three questions on the ballot and asks voters to indicate their first choice and, if they wish, a second one. If one option receives 50% of the votes, it wins. Otherwise, the question receiving the fewest first preferences is eliminated and its proponents’ second choices are distributed over the remaining options. The one that has most votes after this is the winner.

(3) A Condorcet vote, named after an eighteenth-century French philosopher, consists of a single ballot that runs three, two-way races: A vs B, B vs C and C vs A.

If we have three options on the ballot, there are six possible orderings that an individual might have. These are given in Block (I) of the table and for concreteness, each is labelled with the name of a Conservative minister whose words or actions suggest that they would adhere to that ordering. Thus, for example, John Redwood, having long opposed the EU, clearly prefers ‘no deal’ to ‘deal’; Ken Clarke, on the other hand, would like to ‘remain’ but accepts ‘deal’ as the politically responsible thing to do. Boris Johnson and Justine Greening have both declared the PM’s deal to be the worst of both worlds.

Block (II) of the table gives four entirely fictitious distributions of voters (in percent) across orderings; it is not intended to imply any judgement about actual preferences in the UK.

Distribution 1 has the electorate gravitating towards ‘deal’ as a compromise (Mrs May’s dream outcome). 37 percent prefer ‘deal’ (Foxes + Clarkes) to 32 prefer ‘no deal’ (Redwoods + Johnsons) and 31 prefer ‘remain’ (Greenings + Soubreys). ‘Deal’ wins in all three vote formats.

Distribution 2 is much more extreme, with 45 percent preferring ‘no deal’ to 10 for ‘deal’ and 45 for ‘remain’.

Distribution 3 has ‘remain’ as the favoured single option, but still with not enough preferences to beat the combined ‘leave’ positions.

Distribution 4 just shows that gaming is possible in the alternative vote. It is distribution 3 except that 5 Soubreys pretend to be Johnsons. Now ‘deal’ has fewest first preferences (30) and is eliminated and, after redistribution, ‘remain’ beats ‘no deal’ by 62 to 38. But, of course, distribution 4 could arise from genuine preferences, in which case we see that different rules for the three-way ballot imply different outcomes.

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