A recent YouGov poll last week, Tuesday 30 May, provoked a discussion of a likely ‘hung parliament’ in the upcoming UK election. Voters are challenged, faced with many candidates across a spectrum of issues, as the old left-right dichotomy breaks down.
In the literature on voting theory, there is a phenomenon called the Condorcet Loser. If there is a candidate that loses in a 1:1 comparison to each of the other candidates, then that candidate should not be the winner of an election. But it could be the winner. Often ignored in the literature as the Cinderella of the election forecasting ball, a Condorcet Loser candidate could win if a unique set of circumstances prevail. One such circumstance is the voting rule. The UK election will be decided on plurality voting. Plurality voting, known as first-past-the post (FPTP), violates the Condorcet Loser criterion. In other words, there is the probability that in some constituencies a Condorcet loser candidate could emerge as the winner under FPTP.
If a sufficient number of Condorcet Loser candidates are Labour candidates, and if rational voters re-order their preferences along the lines of our stylised example below, then Labour could win the election. How is it possible? Yes, anything is possible. There are many candidates across many constituencies in the upcoming election.
Lexicographic Voter Preferences
Lexicographic preferences exist when different voters have different orderings in their voting preferences such that different voters rank candidates in different ways. In economics, you may choose a bag with less fruit because it has more apples than oranges. This is the win set. So, for example, a Labour/Leave voter might rank a Conservative Brexiteer candidate higher than a Labour candidate or a Conservative/Remain candidate might rank a Green party candidate higher than a Conservative candidate. The more traditional left-right win set is created when candidates’ promises are in sync with left-right voter preferences. To get elected, a candidate, has to position their manifesto in a win set. So, a right-wing candidate promises tax policies and a left-wing candidate promises welfare funding. However, a distinctive feature of this election is that the traditional left-right has been displaced across the spectrum by leadership style and integrity v trust and security, coupled with Brexit. It is neither left nor right but lexicographic: as the candidates send signals to recipient voters it is possible that a Labour/Leave voter might rank a Conservative Brexiteer candidate higher than a Labour candidate because of trust and security or a Conservative/Remain candidate might rank a Green party candidate higher than a Conservative candidate because of leadership style and integrity.
These lexicographic preferences translate into discontinuities, a range of factors (assumptions), if you will, that could facilitate a Condorcet Loser in this election:
- Faced with a range of issues such as social care, taxation, jobs, pensions inter alia, the win set is boundless.
- With so many issues there is the risk of independence of irrelevant (boundless) alternatives facing the electorate.
- Polls for or against a candidate, whether leave or remain are irrelevant, if they display a latent collective irrationality of voter opposition to a candidate, that is, independent of any political party, Brexiteers will always vote for Leave candidate, Labour or Conservative.
- Brexit has endogenized the household balance sheet into the voter preference as rational voters discount their future incomes as argued in the Blog on wistful economics.
If we add in other political parties and a disparity in age-specific voter turnout ratios then it is probable that the UK election process could create a Condorcet Loser candidate in some constituencies.
What if? Stylised Example
A fuller discussion of voting paradoxes in the economics of public choice can be found in both Mueller (2003) and McNutt (2002). Here, by way of a stylised example adapted for the UK election, we assume that the Condorcet Loser is Labour/Leave (LL). The Condorcet Loser is LL since every other candidate is preferred by 2 out of 3 voters in this example. Using the convention in the voting literature of representing preference by the > symbol, each line reads as the voter’s preference so the ordering in line (i) reads as follows: first preference is a Conservative Leave (CL) candidate, second preference is a Conservative (C) candidate, third preference is a Labour Leave (LL) candidate and least preferred is a Labour (L) candidate.
- CL > C > LL > L
- C > L > LL > CL
- L > CL > LL > C
Based on the assumptions 1 to 4 earlier, we place ourselves in the mind of the rational voter with lexicographic preferences on election day. We take a representative sample of rational voters who discard their least preferred choice based on their linear preferences as follows (i) Conservative ‘Brexiteer’ (CL) would never vote for a Labour (L) candidate; (ii) Remain voters would never vote for a Leave candidate, independent of political party and (iii) Corbynites would never vote Conservative (C).
If these conditions held in the mind of the voter on election day then we have a re-ordering as follows:
- CL > C > LL
- C > L > LL
- L > CL > LL
In this scenario, LL is the Condorcet Loser but not a winner. No surprises.
Loser as a Winner Phenomenon
Earlier we had introduced the assumption of ‘leave neutrality’. This could manifest itself in the mind of the voters as they weigh up leadership style and integrity v trust and security into a lexicographic ordering. If the Labour/Leave candidate’s policy platform is minimally differentiated from a Conservative/Leave candidate then a rational Leave voter could rate both candidates nearly equal but ranks Labour/Leave over Conservative/Leave.
If they are equally likely the following re-ordering could happen in the mind of the voter with lexicographic preferences:
- CL > LL > C
- C > LL > CL
- CL > C > LL
By the Condorcet winner criterion, CL > LL by 2:1 and C > LL by 2:1 implying that LL is the Condorcet Loser.
However, with leave neutrality, assumption 2 and the possibility theorem, if the rational voter least preferring LL also rates CL and C low, and if sufficient other voters rate LL close to their preference in the win set, LL could be elected as a majority Condorcet Loser winner. And a Labour candidate takes the seat. So, it goes.
An answer depends on three voting rule requirements: Leave Neutrality: a rational Leave voter rates Conservative and Labour candidates nearly equal but ranks one over the other. Independence of Irrelevant Alternatives: the choice of a candidate in some constituencies is independent of Remain/Leave. Possibility Theorem: the possibility of voters with lexical preferences orderings across n ≤ 5 candidates. Adaptation of McNutt (1993): ‘A Note on Calculating Condorcet Probabilities’, Public Choice vol 75 pp357-361.
 Denis Mueller (2003): Public Choice III Cambridge University Press, Patrick McNutt (2002): The Economics of Public Choice, Elgar Publishing, UK.