{"id":3079,"date":"2019-01-14T09:33:02","date_gmt":"2019-01-14T09:33:02","guid":{"rendered":"http:\/\/blogs.sussex.ac.uk\/uktpo\/?p=3079"},"modified":"2019-01-28T09:17:26","modified_gmt":"2019-01-28T09:17:26","slug":"three-way-voting-paradoxes","status":"publish","type":"post","link":"https:\/\/blogs.sussex.ac.uk\/uktpo\/2019\/01\/14\/three-way-voting-paradoxes\/","title":{"rendered":"Three-way voting paradoxes"},"content":{"rendered":"Share this article: <\/em> \"Facebook\"<\/a>\"twitter\"<\/a>\"reddit\"<\/a>\"pinterest\"<\/a>\"linkedin\"<\/a>\"mail\"<\/a>

\"Image16 January 2019<\/em><\/p>\n

L. Alan Winters\u00a0CB, Professor of Economics and Director of the UK Trade Policy Observatory<\/em><\/p>\n

This note supplements an article on \u2018Organising a three-way referendum\u2019 published on\u00a0The Economist<\/em><\/a> website (16th<\/sup> January 2019).\u00a0It 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\u2019s result that there is no ideal way of combining individual preferences to select one of three options.<\/p>\n

I consider three ways of conducting a three-way vote:<\/strong><\/p>\n

(1) A two-round vote<\/strong>, which would ask in:<\/p>\n

Round 1: \u201cIf<\/em><\/strong>\u00a0 we leave the EU, would you prefer \u2018no deal\u2019 or \u2018the prime minister\u2019s deal\u2019?\u201d<\/p>\n

And then, when the answer is known, to ask (say, a month later to allow time for campaigning):<\/p>\n

Round 2: \u201cNow you know how<\/em><\/strong> we would leave, would you prefer\u00a0to leave or to remain?\u201d<\/p>\n

This ordering solves the fundamental flaw of the referendum in June 2016\u2014that no-one knew what \u201cleave\u201d\u2019 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.<\/p>\n

(2) An alternative vote, <\/strong>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\u2019 second choices are distributed over the remaining options. The one that has most votes after this is the winner.<\/p>\n

(3) A Condorcet vote<\/strong>, 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.<\/p>\n

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 \u2018no deal\u2019 to \u2018deal\u2019; Ken Clarke, on the other hand, would like to \u2018remain\u2019 but accepts \u2018deal\u2019 as the politically responsible thing to do. Boris Johnson and Justine Greening have both declared the PM\u2019s deal to be the worst of both worlds.<\/p>\n

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.<\/p>\n

\"\"<\/p>\n

Distribution 1<\/strong> has the electorate gravitating towards \u2018deal\u2019 as a compromise (Mrs May\u2019s dream outcome). 37 percent prefer \u2018deal\u2019 (Foxes + Clarkes) to 32 prefer \u2018no deal\u2019 (Redwoods + Johnsons) and 31 prefer \u2018remain\u2019 (Greenings + Soubreys). \u2018Deal\u2019 wins in all three vote formats.<\/p>\n

\"\"<\/p>\n

Distribution 2<\/strong> is much more extreme, with 45 percent preferring \u2018no deal\u2019 to 10 for \u2018deal\u2019 and 45 for \u2018remain\u2019.<\/p>\n

\"\"<\/p>\n

Distribution 3<\/strong> has \u2018remain\u2019 as the favoured single option, but still with not enough preferences to beat the combined \u2018leave\u2019 positions.<\/p>\n

\"\"<\/p>\n

Distribution 4<\/strong> just shows that gaming is possible in the alternative vote. It is distribution 3 except that 5 Soubreys pretend to be Johnsons. Now \u2018deal\u2019 has fewest first preferences (30) and is eliminated and, after redistribution, \u2018remain\u2019 beats \u2018no deal\u2019 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.<\/p>\n

Disclaimer:<\/em>
\nThe opinions expressed in this blog are those of the author alone and do not necessarily represent the opinions of the University of Sussex or UK Trade Policy Observatory.
\n<\/em><\/p>\n

Republishing guidelines:<\/em>
\nThe UK Trade Policy Observatory\u00a0believes in the free flow of information and encourages readers to cite our materials, providing due acknowledgement.\u00a0For online use, this should be a link to the original resource on our website. We do not publish under a Creative Commons\u00a0license. This means you CANNOT republish our articles online or in print for free.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

Share this article: 16 January 2019 L. Alan Winters\u00a0CB, Professor of Economics and Director of the UK Trade Policy Observatory This note supplements an article on \u2018Organising a three-way referendum\u2019 published on\u00a0The Economist website (16th January 2019).\u00a0It 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\u2019s result that there is no ideal way of combining individual preferences to select one of three options.<\/p>\n","protected":false},"author":213,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[130195],"tags":[147942,147927,123634,147940,147939,147941],"_links":{"self":[{"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/posts\/3079"}],"collection":[{"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/users\/213"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/comments?post=3079"}],"version-history":[{"count":8,"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/posts\/3079\/revisions"}],"predecessor-version":[{"id":3117,"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/posts\/3079\/revisions\/3117"}],"wp:attachment":[{"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/media?parent=3079"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/categories?post=3079"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.sussex.ac.uk\/uktpo\/wp-json\/wp\/v2\/tags?post=3079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}