Friday, October 14, 2005

The Complexity of Majority Rule

No updates for a few days... The start of term has been manic I'm afraid. One thing I've had to do is write a presentation for this morning's class. I'll post it here in case anyone's interested. Needless to say, plagiarism strictly forbidden!

1. Is majority rule a good idea? Why is it more complicated than it looks?

To understand whether majority rule is a good or bad idea, we need to understand it in its full complexity. Therefore I propose to tackle the two questions in reverse order. That is, I begin by raising reasons why majority rule is more complicated than it appears to many casual observers, then (drawing partly on this analysis) argue that it is not a good idea.

When people refer to ‘majority rule’, they generally have in mind democracy (= rule of [all] the people),[1] and the assumption that when the people disagree then the decision of most must pass for that of all. As Locke says:

“[W]hen any number of men have, by the consent of every individual, made a community, they have thereby made that community one body, with a power to act
as one body, which is only by the will and determination of the majority. For that which acts any community, being only the consent of the individuals of it, and it being one body, must move one way, it is necessary the body should move that way whither the greater force carries it, which is the consent of the majority, or else it is impossible it should act or continue one body, one community, which the consent of every individual that united into it agreed that it should ; and so every one is bound by that consent to be concluded by the majority. And therefore we see that in assemblies empowered to act by positive laws which empowers them, the act of the majority passes for the act of the whole, and of course determines as having, by the law of Nature and reason, the power of the whole”[2]
In fact, matters are more complicated than this. For a start, most people fail to distinguish an absolute majority (50%+1), from a relative majority (plurality), which is simply more than any other option – and could be as little as (v/n)%+1.[3] An absolute majority is a more convincing mandate, but is only guaranteed when there are just two options (and an odd number of non-abstaining voters). The British ‘first past the post’ system elects MPs on the basis of just a relative majority, e.g. in 2005 Clwyd West returned a Conservative MP with just 37.2% of the vote.[4]

The resulting composition of parliament was that Labour won 358 seats with just 35.3% of the national vote, while the Conservatives took almost as many votes – 32.3% - but won only 198 seats.[5] This is by no means an anomaly. In 1874, 1886, 1895 and 1906 UK General Elections produced a landslide of seats for one party that was based on system bias, not popular vote.[6] Nor is it confined to the British case. Abraham Lincoln won a majority in the Electoral College with less than 40% of the popular vote in 1860.[7]

It seems sometimes institutions must be used to create majorities. There may simply be no other obvious way of identifying what ‘the majority’ want, e.g. Condorcet’s majority-cycling.[8] One problem with this, however, is that different institutional rules produce different outcomes. It’s then the institution that determines the outcome, not the preferences of people in society. This isn’t what we allegedly want!

A related problem occurs when there are more than two options. As I said above, an absolute majority is only guaranteed in this case. When there are more options, particularly on two dimensions, the formation of a majority is complicated, and depends on institutions. For example, the extension of the franchise in the UK – it could be extended to all men, to women of the same property qualifications as men, or both. That is, there were four options[9]:

A Limited male franchise; Male only franchise
B Limited male franchise; Give women the vote
C Extend male franchise; Male only franchise
D Extend male franchise; Give women the vote

A majority might prefer D to the status quo (A), but if motions are considered separately for female or wider male franchise they may still lose. If all four options are considered together, matters are even more complicated.

One thing many electoral systems try to do is artificially reduce choices to binary. According to Riker, there is no fair way of doing this. Certainly there is no neutral way – the reason a plurality run-off and sequential run-off produce different ultimate outcomes is that they proceed through the elimination differently. The below example[10] features 55 people, in six groups, and shows how six seemingly reasonable procedures each give different results over their preferences A-E.

I (18 people): A D E _ C B
II (12): B E D _ C A
III (10): C B E D _ A
IV (9): D C E _ B A
V (4): E B D _ C A
VI (2): E C D _ B A

Each group (I-VI) includes the number of people in brackets. The underscore marks the acceptance point – the group accepts all candidates above (to the left of) this. Thus six procedures give different results:

Method: Winner
How it works

Simple plurality: A
A vote is held with all five options. The option with the most votes wins. (This is like our current first-past-the-post system).
A wins, with 18 votes (options B-E scoring 12, 10, 9 and 6 respectively).

Plurality runoff: B
Here a first-vote is taken as above. Instead of declaring a winner at this stage, the two with most votes enter a second-ballot. Now there are only two options, one should win a majority – though voters won’t necessarily be voting for their first-preference. (This is like French presidential elections).
Here, we see A and B make it to the second round. Here, however B has 37 votes (12+10+9+4+2), beating A’s 18 – so B wins.

Sequential runoff: C
A vote is taken of all five. The least popular (e.g. E in the first-round) is eliminated. The process is repeated with four options, again eliminating the least popular. This is continued until there are two options – the least popular is eliminated and the other wins.
E is eliminated in the first round with just 6 votes. These are redistributed to B and C (the second-preferences of those voters). Thus D goes at the next round, and those votes are also redistributed to C. At the third-vote, A has 18, B 16, and C 21. Thus B goes. The final vote is between A and C, and C wins 37 to 18.

Borda count: D
As described above, each gets four points per first-preference, three points per second-preference… to zero points for a fifth (last) preference.
D gets 136 points = (18x3)+(12x2)+(10x1)+(9x4)+(4x2)+(2x2). This wins. A gets 72, B 101, C 107 and E 134.

Condorcet: E
The Condorcet method identifies if one option is a ‘Condorcet winner’, that is, beats every other option in pairwise comparison. There are fifteen ways of pairing five options. Each option is therefore pitted against each of the other four individually. If any option wins each of those four comparisons, it is declared the winner. (This method has nothing to say if there is no Condorcet winner – e.g. in the majority-cycling case).
Here E wins. E beats A 37-to-18, B 33-to-22, C 36-to-19, and D 28-to-27.

Approval voting: D and E tie
In this system, everyone casts a vote for each option they consider ‘acceptable’. They can have as many or as few votes as they like. The option ‘approved’ by the most people wins. (Note: if everyone ‘approves’ only their first-preference, this is equivalent to plurality-vote).
The empty cells above illustrate the ‘approval threshold’. All the voters approve their top three preferences, except group III (who approve their top four). A is approved by 18, B by 26, C by 21, and D and E by all 55. Thus D and E tie.

What we see here is that, even given full information about voters’ preferences, it’s not obvious which option should be declared the ‘winner’. Note the procedures here aren’t, of course, simple majority-rule. None of the five options commands a majority (at first). Although the Borda count and approval voting aren’t directly related to majority-rule, the other methods are all attempts at capturing its essence. The plurality-vote is a weaker approximation of majority-rule, requiring only a ‘relative’ majority. The plurality and sequential run-off methods are both ways of whittling options down to two, so majority-rule becomes applicable. The Condorcet method employs majority-rule in a series of two option comparisons.

The ‘majority’ that rules is, it seems, largely arbitrary and meaningless. Moreover, it is not just that one possible majority is selected (at random) by the system, instead the system ‘creates’ majorities where none existed before. For example, by narrowing a four-way contest down to a two-horse race, it means one party must have a majority, but this doesn’t mean they have the real support of 50%+1 – as many of those would really have preferred another candidate, and hence another electoral system that gave their preference a chance.

Representative systems typically illustrate the creation of majorities in another way. A majority of representatives need not represent the majority of the larger population. In theory, 50%+1 of the MPs could’ve each been narrowly elected in their constituency – by just 50%+1 of voters, in each of those 50%+1 constituencies – and thus speak for little over a quarter of the population. The other almost 75% of the population could be against a motion passed by the majority of MPs![11] Which majority should rule?

If majority rule seems meaningless, then it’s hard to argue that it is a good idea. Epistemic democrats (e.g. Rousseau, Estlund) argue the majority are our most reliable guide to the ‘right’ outcome. This not only assumes one uniquely right outcome – controversial in modern heterogeneous democracies – but also that voters reach independent judgements and have a greater than 0.5 chance of being individually right on any particular issue. These assumptions are, if not downright unrealistic, controversial.

A utilitarian justification for majority rule is likely to proceed in terms of it being ‘better’ – which need not make the assumption it is better for everyone (like Rousseau’s General Will), it can appeal simply to the net sum of satisfactions. As Sen has argued, however, preference-satisfaction is an inadequate measure of well-being.[12] In any case, there is not enough information revealed in voting to guarantee that utilitarian optima are reached. Majority rule, for example, allows an almost indifferent majority to prevail over an intense minority, when the reverse would bring about greater net satisfaction.[13]

Even if we assume that people are better off if they get their way, and that more people getting their way is socially better (i.e. a greater net satisfaction), repeated majority rule may produce repeated winners and losers, which may violate notions of fairness. For example, in a district split 51/49 (in some permanent way, e.g. along race lines), the 51% will get their way 100% of the time, which hardly seems fair. They could, for example, decide to enslave the others.[14] Even if a new vote was taken each week on who were to be slaves and who masters, the 51% would win every time!

Sometimes majority rule may seem natural and appropriate; for example, if there is a one-off binary decision to be made where each has an equal stake. It certainly does not guarantee right, good or fair outcomes, however – probably no procedure can. Moreover it is dogged by problems of defining ‘the majority’, which is often more an artefact of the electoral system, rather than a real reflection of people’s preferences. As such, it struggles when there are more than two options or one dimension of issues, as revealed by majority cycling. One cannot assume majority rule is a good idea in all contexts.

Thankfully there are alternatives to majority rule that treat each equally. The appeal of lottery-voting was illustrated to me by an actual case arising in our GCR – and I’m certainly not the first to be stimulated by college politics.[15] In Trinity 2004, there was a proposal to buy some reference books,[16] but there were a range of views on what to do – some wanted top-of-the-range books costing £200, others to reject the motion altogether and most in between – a reference library but at a lower budget. The outcome seems to depend more on decision mechanism than preferences. E.g. one might write down how much everyone would spend and take an average (presumably more than £0 but less than £200). Or set several options (£0, £50, £100, £200) and have people vote which. Even that could use simple (relative) majority or some means of alternative voting (like our GCR elections). Or have a yes/no vote followed, if applicable, by decision over how much to spend (perhaps simply a high/low choice) – or the reverse (set a hypothetical amount, then decide whether to spend it).[17]

In these situations, any ‘majority decision’ will either fail or succeed only due to the voting mechanism. There’s a case for taking an average, which minimises aggregate dissatisfaction (how far each voter’s preference is from the outcome). This makes majorities always hostage to minorities; if 99 people want nothing spent, but one person £200 they still have to spend something (£2). Strategic voting presents another problem – if one wanted £50 but thought most people inclining to £0, one might vote £200 thereby bringing the average artificially closer to one’s preference.[18] Finally, averaging won’t work in cases where what matters more is polarity (which side of the issue one falls on) rather than how far from the middle one is – e.g. the abortion debate (a moderate pro-choicer may have more in common with a radical than a moderate pro-life campaigner)[19]

While acknowledging other procedures may produce a better outcome, I think fairness is satisfied by a ‘random dictator’ model – everyone writes down how much they would spend between £0 and £200 and one choice is selected at random as group policy. This gives everyone an equal chance and respects numbers – if more people favour a particular option (say £100) that becomes more likely – and not just numbers who exactly agree, but those who are close (e.g. someone who wanted £90 isn’t so far from £100).

Of course, this proposal has its own problems – e.g. one outlying voter may be drawn, and the result obviously depends on the institution (which vote is actually drawn) rather than just votes – but each option’s chances are improved by each vote they receive.

[1] As Aristotle noted, this isn’t necessary. Even oligarchies operate by majority rule, so what distinguishes democracy is the franchise.
[2] Locke Second Treatise of Government para.96. Various editions, e.g. Two Treatises on Civil Government with an introduction by Henry Morley (London: George Routledge and Sons, 1884) p.241. I am not the first commentator to find this unconvincing. See, e.g. Gough (1950) John Locke’s Political Philosophy pp.60-63 and Simmons (1993) On the Edge of Anarchy: Locke, Consent and the Limits of Society p.92ff.
[3] v = voters, n = options. If there are 100 voters and four options, 26 votes could be a ‘relative majority’ (if the others are split 25, 25, 24).
[4] Labour 35.9%, Lib Dem 13.3%, Plaid Cymru 10.9%. Source.
Note this also neglects the further complication that turnout was only 64%. Even an absolute majority of those that vote need not be even a relative majority if everyone voted. This is a problem if abstention rates differ between those of different political views, as is likely given that members of poor, disadvantaged minorities are generally less likely to vote.
[5] Source.
[6] McLean (2001) Rational Choice & British Politics p.91.
[7] McLean (2001) Rational Choice & British Politics p.13ff; c.f. McLean (2002) ‘Review Article: William H. Riker and the invention of Heresthetic(s)’ British Journal of Political Science 32 pp.550-3. Note this is complicated by the two-stages of indirectness, which I come to later.
[8] For example, if three people rank options a>b>c, b>c>a and c>a>b then one a first-preference count each option gets one vote, and in pairwise comparison a beats b (2-to-1), b beats c and yet c beats a. The usual solution is a status-quo bias, which is non-neutral and effectively makes one person a dictator (if c is the status quo, person three gets what he wants, though everyone else prefers b).
[9] From McLean (2001) Rational Choice & British Politics p.106.
[10] Originally owing to Malkevitch; reproduced from Shepsle & Bonchek (1997) Analyzing Politics: Rationality, behavior and institutions pp.167-72.
[11] For example, suppose there are five districts, each of 20,000 voters, and each represented by one legislator. Even though there is not one majority-bloc, it may still be that a majority within three of the districts – who are not a majority overall – are sufficient to pass bills affecting the whole polity. It takes only three of the five legislative votes to enact policy. Each of these three legislators could’ve been elected by a narrow majority, however – say, just 12,000 of their 20,000 constituents. In this case, 36,000 voters strategically placed effectively determine policy for the whole 100,000. The other 64,000 may all be opposed – but they win just two districts with over-whelming majorities (indeed, a unanimous 20,000 votes) and lose the crucial three 8,000-to-12,000. This effectively violates anonymity. See Still (1981) ‘Political Equality and Election Systems’ Ethics 91:3 [Special issue: Symposium on the Theory and Practice of Representation] p.382.
[12] His famous examples include a contented slave and a housewife who accepts repression. The danger here is not an outside influence prevailing over their preferences, but shaping their preferences.
[13] The Borda count is an attempt to consider intensity of preferences, but even this fails as it uses an ordinal ranking rather than a cardinal one. Consider the case where there are five options for the income tax rate – 65%, 64%, 63%, 62% and 30%. Someone who ranks them in descending order (65-30) is presumably much more satisfied by his ‘second preference’ than someone who ranks them in the reverse order (whose ideal is 30%, but whose ‘second preference’ is 62%).
[14] Of course, one who thinks there are values prior to the democratic process – as one must, I think – can always think a majority wrong, as with any other decision procedure. Perhaps constitutional protections can be used to block some infringements of a minority’s rights, but the problem persists in essence even if what is decided is something less important – like a number of economic decisions that all benefit the majority.
[15] So was Dodgson in Christ Church – see McLean and Urken (1995) Classics of Social Choice p.52. Many others have come from similar small groups, e.g. the Paris Academy or church groups.
[16] Lengthy debate was involved in the first meeting of Trinity Term 2004 before a rather vague motion was thrown out. A refined motion was subsequently passed in the second meeting of term. This discussion draws on my communication with the GCR president, particularly an email of 28th May 2004.
[17] If we decide to buy books, then settle the amount, presumably many who’d been opposed to the purchase would then support buying the cheapest books. This is not guaranteed however – some may prefer we didn’t buy books, but if we were going to we ought to get good ones. (This is one factor complicating the ‘aggregation’). If we decide on the amount first, it’s perhaps more likely those opposed to any purchase would go for a higher figure, if they think this will swing other voters (who want cheaper books) into voting against the purchase altogether. (Again, they might in fact decide on a lower price, if they think that will be more effective in encouraging people to vote against the purchase – the majority believing cheap books aren’t worth having, for example).
[18] I owe this point to Toby Ord. One of the advantages of lottery voting is that it encourages true preference reporting.
[19] We can’t assume averaging best in the book example either. £2 would only buy a very cheap dictionary – those who opposed buying anything may see that as literally ‘worse than nothing’ and prefer that if we bought anything it should be of reasonable quality.


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