The Brexit process is marooned in uncertainty as no democratic method seems able to select a route that is favoured by a parliamentary majority. The problem is not new, however, and game theory offers a solution writes John Egan.
Last night, the UK parliament held its long awaited sequence of indicative votes to select a possible route to Brexit which could command majority support. Eight widely different options were selected to cover the range of possibilities – although these did not include the form of Brexit negotiated by Prime Minister Theresa May that had already suffered two brutal defeats in earlier Meaningful Votes.
On specially prepared pink voting slips MPs indicated “yes”, “no” or left blank their preference for each of the eight selected options. The combined democratic outcome was eight “no” rejections – although one proposal for a “permanent and comprehensive UK-wide customs union with the EU” by Father of the House Kenneth Clark failed by only eight votes.
This result should have been expected. Each proposed option suffered the same fate as Theresa May’s deal. In each case the multiple alternatives NOT to choose that option simply had more combined support.
However, there is likely to be one option that does command majority support in a head-to-head run off with all other alternatives. This is called the Condorcet Winner, named after Nicolas de Condorcet, a polymath and major lumière of the French enlightenment.
‘With more than two alternatives and strategic voting, standard one-shot voting procedures, such as first past the post (plurality) or single transferrable voting, cannot ensure that the alternative that is favoured by a majority is chosen.’ Toke Aidt, University of Cambridge
The key challenge is to devise a system to find the Condorcet Winner and three senor economists and game theory experts from Cambridge and London published a paper in February with a system devised specifically to deal with the democratic impasse in which Brexit is to be found .
Condorcet solved the problem himself by requiring all voters to list their preferences in numerical order from the most to the least preferred. These lists can be used to indicate the winners of every possible head-to-head play off, and the option with the most of these winners is actually the Condorcet Winner .
However, there is a problem. Condorcet was an unrepentant idealist and optimist on the virtues of human nature. It was a reputation that led to his death in the aftermath of the French revolution. His method works, but only if the voters choose their preferences sincerely according to their beliefs. When voters behave strategically, for example by choosing options that block others or perhaps that lead to prime ministerial resignations, Condorcet’s method may fail to select the Condorcet Winner.
It is not unrealistic to expect that some MPs could vote strategically to some considerable degree.
To select the Condorcet Winner in the case of strategic voting a “Weakest Link” method is proposed whereby a series of votes sequentially eliminates the option that is favoured the least until there is one left standing – the Condorcet Winner . Among many other applications, this approach is used to select the leader of the Conservative Party, where MPs vote until the final two remaining candidates face a wider electorate of all party members.
The Weakest Link has its own weakness. It may fail to select the Condorcet Winner when voters behave sincerely.
To deal with a combination of sincere and strategic voting, the paper published in February proposes a combination of the Condorcet and Weakest Link methods: “As in the Weakest Link, decisions take place sequentially and in each round one option is eliminated. However, instead of eliminating the option with the least votes, the Condorcet method is applied in each round and the option that loses to most other options is eliminated.”
The authors assert that this “Sequential Condorcet” method ensures that the Condorcet Winner is selected unless voters’ preferences are systematically related to whether they vote strategically or sincerely. It is also possible that a Condorcet Winner does not exist, being replaced by a circular outcome where option A beats B, which beats C, which in turn beats A. However, this is considered unlikely for Brexit.
In their paper, the Cambridge and London authors refine their proposed methodology to remove potential bias by insisting a full range of options should be considered and each should be treated in exactly the same way.
One of the authors, Toke Aidt from the University of Cambridge, concludes: “We believe that there is a strong case for adopting our procedure to resolve the Brexit impasse in the House of Commons. It would have to be undertaken in a sequence of indicative votes that would end with a binding vote between the two last alternatives. With more than two alternatives and strategic voting, standard one-shot voting procedures, such as first past the post (plurality) or single transferrable voting, cannot ensure that the alternative that is favoured by a majority is chosen.”
It is this complexity that brings into focus the weaknesses introduced by the initial 2016 Brexit referendum and the parliamentary procedure that followed. A complexity which may be inherited by any subsequent People’s Vote.
 Breaking the Brexit Impasse: Achieving a Fair, Legitimate and Democratic Outcome by Toke Aidt, Jagjit S. Chadha and Hamid Sabourian. National Institute Economic Review No. 247 February 2019.
 This Condorcet Winner is the option that wins n-1 ballots, where n is the number of options
 It is assumed that there is complete information about the preferences of different voters and a complex backwards induction reasoning is required of voters. In practice, it is recognised that this information will be incomplete.