Written evidence submitted by Dr Ben Saunders (VPC 02)

Biography

I am Dr Ben Saunders, Associate Professor in the department of Politics and International Relations at the University of Southampton. I have a doctorate in politics from the University of Oxford. I teach and conduct research into democratic procedures, including voting systems. I am responding in a personal capacity and do not claim to represent my employer or any other organisation.

Executive Summary

* Voting between only two options is relatively straightforward, but cases involving three or more options raise a number of problems.

* Seeking to decompose a three-way choice into a series of binary choices may produce paradoxical results (such as majority cycling).

* Attempts to reduce three or more options to two can also produce undesirable results, such as non-monotonicity (where voting for one option may prevent it from winning).

* Approval voting allows each voter (MP) to express which option(s) they find acceptable.

* This amounts to something similar to the series of ‘indicative votes’ taken earlier this year, although under approval voting majority support is not required.

Full Response

[1] Votes, or divisions, within the House are currently over binary options, ‘aye’ and ‘no’. When choosing between two options, simple majority rule has been proven to satisfy a number of generally attractive conditions: decisiveness, neutrality (no option is favoured), anonymity (no voter is privileged), and non-negative responsiveness (May, 1952). However, despite the attractions of this voting method, there are difficulties when confronted with more than two options.

[2] When considering more than two options, a relative majority or ‘plurality’-based system may deliver outcomes that a majority of voters (where ‘voters’ here refers to MPs) dislike.[1] Consider a simplified case involving three options (A, B, and C) and ten voters. We might let each voter vote for their most preferred option and declare the option with the most votes the winner. (This is how ‘first past the post’ elections work.)

[3] One trouble with this is that a majority of voters may prefer either of the defeated options to the winner, as illustrated below:

Here A has four votes, while B and C have three votes each. Thus, A is declared the winner. However, six of the ten voters prefer B to A. And six of the ten prefer C to A. Since a majority of voters prefer either of the other options, it seems unsatisfactory to declare A the group’s choice.

[4] It should also be obvious that this procedure is susceptible to manipulation, by strategic voting or adding/removing options. For instance, if B were to drop out of the contest, then C would win. It is widely considered unsatisfactory for the group’s choice between A and C to depend on whether or not B is an option.

[5] One solution to this problem is to consider options pairwise. This is how the House currently votes. First, there will be a vote between A and B, which (in the example above) B wins. Then a vote between B and C, which B wins. However, while a succession of pairwise votes avoids certain problems, it can also create others.

[6] One difficulty with pairwise voting is that there may be a Condorcet paradox or cycle, that is a case where no option enjoys a stable majority (this is also known as a case of intransitivity). This is illustrated below, in a case involving only three voters:

Here, D beats E by two votes (Voters 1 and 3) to one. But D then loses to F, again by two votes (Voters 2 and 3) to one. Thus, we might conclude that F is the group’s choice, for it beat D which, in turn, had already defeated E. However, if we hold a vote between E and F, then E would win by two votes (Voters 1 and 2) to one. Thus, E > F > D > E > F > D, etc.

[7] It is difficult to estimate how frequently such paradoxes occur in practice, since a third vote is hardly ever held. In the above example, F would ordinarily be declared the winner after the first two votes. However, this means that the outcome depends on the order in which options were voted on. Voting between D and E, then D and F produces F as the winner. But had the first vote been between E and F, then E would have won, the second vote would have been between D and E, and D would be declared the winner.

[8] The relevance of this for House of Commons procedures should be evident if we imagine that D = the current status quo, E = a proposed bill for change, and F = a possible amendment to that bill. It may be that a majority of the House prefer the original bill (E) to a proposed amendment (F), so the amendment fails. And a majority of the House prefers the status quo (D) to the original bill (E), so the bill is rejected. But, had there been a vote between the status quo (D) and the amended bill (F), then a majority would have supported the amended bill. Thus, the outcome of the House’s deliberations depends on whether it (i) first decides between the bill and the amendment and then whether to change the status quo or (ii) first decides whether to change the status quo and then votes between the original bill and its amendment.

[9] Note that these results do not depend on any assumptions about strategic voting or the like. There is no suggestion that, for instance, an amendment has been proposed in order to wreck a bill that may otherwise have passed. While strategic behaviour may also be a problem for designing voting systems, and no non-random, non-dictatorial procedure can avoid the possibility of manipulation (Gibbard, 1973; Satterthwaite, 1975), in these examples all voters are assumed to vote according to their sincere preferences.

[10] This problem, along with Kenneth Arrow’s (1963) generalisation of it and more recent work on the so-called discursive dilemma (Pettit, 2001),[2] has led a number of scholars of social choice, such as William H. Riker, to be sceptical of any claim or assumption that a group decision reflects any ‘group will’ (Ingham, 2019). There is certainly good reason to be cautious in interpreting the results of group decision procedures, but this does not obviate the need for groups to make decisions together.

[11] While these problems can be summarised simply as showing that no voting procedure is perfect, a number of procedures have been devised, some of which are better or worse for particular purposes. It is important to understand the features—and the potential problems—of whatever voting procedure is adopted. This is particularly the case for the weighty business of the House of Commons.

[12] Since the problems of social choice only kick in when there are three or more options, one common approach to avoiding them is to winnow the options down until (if necessary) only two remain. One way of doing this is through a sequential run-off, in which the least popular option is eliminated and votes for that option redistributed, until one option has an absolute majority. (This proposal is effectively the ‘Alternative Vote’ that was put to a referendum in 2011, and has similarities to the method used for electing Conservative Party leaders, though there the voters in the final round [party members] are different from the voters in earlier rounds [MPs]. Note, however, that the House using it to decide on legislation may differ from constituencies using it to elect MPs.)

[13] Suppose we have one hundred voters and three options (G, H, and I), with preferences as follows:

Here, in the first round of voting, G has 39 votes, H has 35 votes, and I has 26 votes, so I is eliminated. These votes are transferred to G, so G wins in the second round, by 65 to 35.

[14] However, suppose the ten voters in the second row change their preferences, so they now rank G first. No other voter preferences change. Voter preferences now look like this:

In the first round, G has 49 votes, H has 25 votes, and I has 26 votes. In this case, H is eliminated and those 25 votes pass to I. The result is that I wins, by 51 votes to 49. Some voters changing their votes in favour of G actually prevented G from winning. This is known as non-monotonicity. (This was a significant reason that the Plant Report did not recommended STV elections.)

[15] Many commentators think that non-monotonicity is perverse (Doron and Kronick, 1977). While it has been estimated that non-monotonicity is very unlikely in STV elections (Allard, 1995 and Bradley, 1995), it cannot be assumed that this would also be true of House votes on legislation. If G, H, and I represented alternative pieces of legislation, in a three-way vote using a run-off method, it would be perturbing to find that a particular bill was defeated, but would have passed had some of those MPs who voted for it switched their votes to another option.

[16] Another option for dealing with multiple option is approval voting (Brams and Fishburn, 1978). Voters are presented with all possible options and may vote for (approve of) as many or as few as they like. The option approved of by most voters, whether or not a majority, wins. This is, effectively, how a Doodle poll operates. Those invited are given a range of meeting times and select all those that are satisfactory.

[17] Approval voting may seem to depart from ‘one person, one vote’ since, if presented with seven options, one voter may vote for two while another voter may vote for five. However, this does not amount to an objectionable form of inequality. Each voter has the same power and each vote carries the same weight. We might think of what’s going on as seven separate yes/no votes taking place at the same time, to see which of the seven options receives most approval.

[18] The series of ‘indicative votes’ that the House took on Brexit proposals were, therefore, rather like a round of approval voting, except that approval voting would not require majority approval for any particular option. Under approval voting, an option may be declared the winner, even if it does not have majority approval. While it is preferable that decisions are widely approved of, at least this prevents gridlock or the possibility of greatly disliked outcomes. The indicative votes revealed that the House was strongly against a No Deal Brexit, though that is what will happen, unless some alternative is agreed. Thus, even though various alternatives failed to win majority support, it is likely that many of them would be preferred to No Deal.

[19] This is not to say that approval voting is without any faults. First, it is still subject to strategic voting. That is, some voters may pretend not to approve of options that they actually consider acceptable, because they fear that those options are likely to win, at the expense of those voters’ most preferred options. There is some incentive to do this, since voting for second or third choices can harm the prospects of your first choice winning (unlike in a run-off, where one’s second choices would only come into play after one’s first choice is eliminated). There is also an incentive for two similar candidates, or the proposers of two similar options, to encourage their supporters to combine in public, while encouraging defection in private. This gives rise to the so-called Burr dilemma (Nagel, 2007). However, since any voting procedure that is not unacceptable on other grounds may be manipulated (see paragraph 9, above), this is not reason to reject approval voting.

[20] Second, approval voting only permits voters to say yes or no to each option, so it does not register the strength of their approval (or support) for any particular option. An option that is strongly favoured by one voter is treated equally to one only grudgingly approved of by another; both receive one vote. Thus, an option that is widely regarded as (merely) ‘ok’ may defeat one that has the passionate support of almost as many people.

[21] However, comparing the strength of one voter’s preferences to those of another is practically impossible. Though some procedures, such as the Borda count (Saari, 1990), attempt to do something like this, by assigning points according to each voter’s preference ordering, the results that they produce are largely arbitrary and very susceptible to manipulation (Risse, 2005).

[22] Approval voting is a popular decision-making mechanism amongst scholars of social choice. It is used by a number of learned societies, including to elect the president and council of the Society for Social Choice and Welfare (see article 5(4) and article 6(3) of their statutes). Since the ‘indicative votes’ recently taken in the House are, in effect, a similar procedure, this suggests that they have some promise for resolving questions where there are more than two options.

[23] The chief difficulty, for the indicative vote process, seemed to be an expectation that one option should command majority support. Where there are numerous options on the table, there is no guarantee that any one of them will command the support of a majority of voters (MPs). However, as demonstrated above, attempts to find a majority in favour of one option are fraught with difficulty. Different procedures may produce different outcomes, with different majorities, suggesting that the winner is not really down to voter preferences, but down to the voting system employed.

[24] I hope that this evidence is of use to the committee’s inquiry. I would be happy to provide further evidence or commentary upon request.

September 2019

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