Written evidence submitted by Toke Aidt, Jagjit S. Chadha and Hamid Sabourian[1]
This submission is a response to the call for evidence on alternative vote procedures when the House of Commons needs to decide amongst more than two options. A more detailed exposition of its logic can be found in the commentary published by the National Institute of Economic and Social Research in February 2019 and in underlying academic research [paper 1].
When the House needs to decide between two well-specified options, the majority rule (aye/nay voting) is the appropriate procedure. In fact, it is only vote procedure, which treats all alternatives and all Members the same and where a switch of one vote from the ayes to the nays (or vice versa) breaks a tie, that will reach a decision. This is call May’s Theorem in the Social Choice literature after Kenneth May who proved this in the 1950s.
The difficulty arises when there are more than two options. In this case, each ballot is a vote on a given proposal against “not that proposal”. For example, if there are three alternatives, A, B and C, then a ballot on A is, in effect, a ballot on A against either B or C. Members may often have different views on which of the two options in the example’s “not that proposal” is desirable and on which is more likely to become a reality if A is voted down. This opens up the real possibility that no proposal can command a majority against “not that proposal” as the sequence of unsuccessful ballots held in 2003 in relation to reform of the House of Lords and the indicative vote in the House of Commons to evaluate several propositions related to the UK’s exit from the EU vividly illustrate. This, however, does not necessarily mean that there is no majority for any proposal. Rather, it illustrates the fundamental weakness of using the majority rule to decide between many alternatives.
Before considering which voting procedure the House might adopt to decide between many options, it is essential to consider which principles the procedure should satisfy. This helps eliminate procedures which may look appealing at first, but which fail on closer investigation.
We take it as given that the Government (or the House) decides the options to vote on using its normal procedures. We suggest that the procedure to select one of these options should satisfy:
First, all options should be treated in the same way. This is the neutrality principle, which ensures fairness. It means, for example, that the order in which the options are considered, does not itself bias the final choice. Violating neutrality would expose the procedure to the accusation that the process is rigged.
Second, the procedure should treat all Members the same way. This is the anonymity principle. The one-person-one vote system ensures this.
Third, the voting procedure should be Condorcet consistent and select the option among all the options considered that commands a majority of votes in head-to-head (pairwise) ballots against all the other options. Such an option is called the Condorcet Winner after 18th century philosopher and mathematician the Marquis de Condorcet. A Condorcet Winner reflects the majority view in the sense that there is no other option that can win a majority vote against it and it encapsulates what is commonly understood by the “majority view”. The Condorcet winner derives its legitimacy from the fact that it is stable, in the sense that once the House selects it there is no other option (among those considered) that can win a majority vote against it. If there is no Condorcet winner, then the voting procedure should not select an option, which fares badly against other alternatives in pairwise votes (we return to this at the end).
In sum, we suggest that the House adopt a vote procedure that treats options and members the same, and which selects the option that can win a majority against all the others considered.
The next question is which voting procedures satisfy these principles. We begin by arguing through an example why applying aye/nay voting to make a decision amongst more than two options is problematic. It violates the principle of Condorcet consistency and it may not reach a conclusion even in cases where there is a Condorcet Winner.
Suppose that there are three options: A, B or C and three voters indexed 1, 2 and 3. Table 1 illustrates their preference ranking over the three options. For example, voter 1 prefers option A to option B to option C and so on for the other voters.
Table 1: Three voters and rank preferences over three options
| Voter 1 | Voter 2 | Voter 3 |
1st | A | B | C |
2nd | B | A | A |
3rd | C | C | B |
Note: The options are listed in each column is declining order of preference.
In this example, option A is the Condorcet Winner because it wins a majority against both option B and C in a pairwise vote (voter 1 and 3 vote A in a ballot between A and B and voter 1 and 2 vote A in a ballot between A and C). We want the vote procedure to select A.
Now, suppose that the voting procedure is to put each proposal to a vote to find a majority in favour of one of them (aye/nay voting). For the sake of argument also suppose that voters vote sincerely, i.e., according to their true preferences for the options (as stated in the table). Option C would clearly fail, as a majority consisting of voter 2 and 3 would vote against because both option A and B are preferred by them to option C. The situation with the other two options is more interesting. Suppose that there is a vote of “option A” against “not option A”. Voter 1 would vote for this because she prefers option A to the other two. However, for voter 2 and 3, the situation is more complicated. “Not option A” is a bundle of options B and C and while each of them prefers one of these options to A, they prefer A to the other. They may, therefore, reasonably vote against option A if they either think that voting A down means that their preferred option (B or C, respectively) is likely to be approved next or if they really value their most-preferred option at lot relative to the rest, or a combination of the two.
In a vote of “option B” against “not option B”, voter 3 would vote yes and voter 2 would vote no; Voters 1 will be decisive but could well vote against option B if her preference for A is sufficiently strong or she thinks A is sufficiently likely to become a reality if B is voted down. In this case, the outcome of the vote procedure would be that there is no “majority for anything”, yet option A is a Condorcet Winner. Alternatively, if voter 1 do value option A that much more than option B, but really dislikes option C and fears that C could become a reality if option B is voted down, then she might instead support option B. Option B, then, wins a majority despite not being the Condorcet winner.
This example demonstrates how problematic it is to use the aye/nay procedure to decide among more than two options. The fundamental problem is the options that are voted on are not well-defined in each ballot – the ‘not that option problem’. The source of the problem is that Members inevitably value alternatives differently and hold subjective beliefs about what might happen if a particular option is voted down. The need for a better procedure, which can, in fact, identify the Condorcet winner and so reach a democratic decision, is clear.
The objective is to devise a vote procedure that can satisfy the three principles of neutrality, anonymity, and Condorcet consistency, and which is simple enough to be practical for votes in the House of Commons. An important consideration is how strategic Members are when they vote. To see why it matters, it is useful to make a distinction between sincere and strategic voting.
Members vote sincerely if they truthfully and myopically vote according to their preferences over the options before them. They do not think strategically about how their vote may interact with those of others and the outcome to which that might lead. They simply consult their preferences over the options presented and vote for the one that they prefer the most. In contrast, Members vote strategically if they do think strategically about how their vote may interact with those of others. They may be willing to vote against an option they, in fact, prefer to another option if they think that strategy will eventually lead to an outcome they like even better.
In reality, Members will, we conjecture, sometimes vote sincerely and sometime strategically and our proposed vote procedure satisfies the three principles for any combination of sincere and strategic voting: it works for all cases. However, to understand what is at stake and why other well-known procedures do not work, we consider, in turn, the extreme cases in which Members vote either sincerely or strategically.
Suppose that Members vote sincerely. It is, then, well-established in the academic social choice literature that a number of commonly used voting procedures may not select the Condorcet winner. This includes the plurality rule (the alternative that gets the most votes win); majority run off (where if an option fails to gain a majority in the first round, then the two options with the highest vote count is voted against each other in the second round); the Borda rule (where each option gets a score according to the ranking in a voter’s preference ordering and the option with the highest score wins) and many others.
However, there is a simple method that works: the Condorcet method. It asks Members to write down their complete ranking of options and then mechanically counts how many pairwise votes each option can win; the alternative that can win K-1 pairwise ballots with K options is the Condorcet Winner. An alternative way to do this is to ask Members to vote on each pair of alternatives. In the example in Table 1, this is straightforward and it only requires three ballots, but if there are many options, the number of ballots is large.
This provides a strong rationale for using the Condorcet method to decide amongst multiple options if Members vote sincerely. However, the simple Condorcet method cannot be guaranteed to select the Condorcet winner if Members vote strategically, so we would not recommend adopting it for voting in the House of Commons.
Given the possibility of strategic voting, an alternative to the Condorcet method, which is robust to that and which satisfies the three principles, is required. Based on underlying research, we suggest a procedure called the Weakest Link. This is a multi-round ballot procedure in which in each round Members vote between all remaining alternatives and the one with the least votes is eliminated. Voting continues until only one option is left. The basic idea can be understood by considering the final round of the procedure. At that point there will be two options, for example, A and B in the example in Table 1. At that point, the best thing for each Member to do is to vote for his or her preferred option, i.e., sincerely. Hence, whichever of the two is preferred by the majority will win the showdown vote in the last round. In the example, option A would win against either B or C because it is the Condorcet Winner, and B would win against C.
If we then work backwards to the previous round when there are three alternatives and it is known to the Members that if a Condorcet Winner reaches the last round (with two options), then it will win against any other option, as shown above. So in this penultimate round the rational strategy for a majority of voters is to make sure that the Condorcet winner is not eliminated and, therefore, to vote for it. In the example in Table 1, voter 1 would vote A because this is his first preference. Voter 3 ideally would like option C, but knows that option cannot win in the final round and she, therefore, strategically votes for A to make sure it is not eliminated. So, by a process of backward induction, we can see that the Weakest Link procedure will get us to the Condorcet Winning if all Members act strategically.
The sequential nature of the procedure is fundamental: any procedure designed with only one round of voting (including the single transferable vote and many other commonly used voting procedures) is insufficient to ensure that the Condorcet winner is selected under strategic voting. The downside of the Weakest Link procedure is that it does not work if Members vote sincerely. In that case, it is possible that the Condorcet winner is eliminated before the final round.
So, when the population of voters consist of a mixture of strategic and sincere voters, which must be considered the most common case, an amended version of the Weakest Link procedure will work and select the Condorcet Winner. This is called the Sequential Condorcet procedure and it is a hybrid of the Condorcet method and the Weakest Link procedure. 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. This ensures that the Condorcet Winner is selected at the end of the process under the assumption that the voters’ preferences over options are not systematically related to whether they vote strategically or sincerely. This is the procedure that we recommend that the House adopts.
We propose that the House adopts the sequential Condorcet procedure in votes with more than two options. The practical implementation of the procedure could work like this:
1) Decide on the K>2 options to be on the ballot.
2) Members rank the options from 1 to K. This can be done on an App so that the rankings can be automatically uploaded to a piece of software that will aid the Tellers.
3) From the rankings, the Tellers establish how many pairwise vote each option can win with a majority and give them a score that is equal to that number. The option with the lowest score (the one that can win the least pairwise votes) is eliminated.
4) Members rank the remaining option from 1 to K-1 and step 3 and 4 is repeated till there is either only one option left or multiple options left with the same score (a tie).
5) If there is only one option left, this is the Condorcet Winner.
6) Tie-breaker rules:
The sequential Condorcet procedure will select the Condorcet Winner from the set of alternatives if there is one. While it is logically possible that no Condorcet Winner exists in a given context, research shows that the problem is most acute for small committees and is least acute when many individuals vote on a few alternatives (the situation in a House of Commons vote). However, even in the absence of a Condorcet Winner, the sequential Condorcet procedure does a good job. It will isolate the set of options which can win a majority against all the options outside that set. This means that the procedure never selects an option that commands minority support in head‐to‐head pairwise voting. It does, of course, leave unanswered the question as to how one then decides between these tied options. We propose, as indicated above, to use the Borda rule to break ties but this is not essential as long as the rule is neutral and treats all options similarly.
September 2019
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[1] Toke Aidt, Reader in Economics University of Cambridge and past-president of the European Public Choice Society. Jagjit S. Chadha, Director, National Institute of Economic and Social Research. Hamid Sabourian, Professor of Economics and Game Theory, Chairman of Faculty of Economics and Director of Keynes Fund, University of Cambridge.