Brexit wrecks it: the theory of collective decision making

Basically, political decision-making should not be win-or-lose, as facilitated by the most ancient, primitive, divisive and inaccurate measure of collective opinion ever invented.

Peter Emerson
28 February 2017

The people have spoken? No no, at 52-48, they have disagreed. Some referendums are OK, like the Belfast Agreement; others are dodgy; a few, as in the Balkans, provoked wars.

Logically, the will of the people cannot be identified if lots of them state only what they do not want. However, if the choice of options is sufficiently broad so that folks can vote positively, we may identify the will of a majority (or maybe just the largest minority). Better still, if they cast their preferences, we can identify the option most acceptable to (almost) everybody.

This article discusses decision-making and makes only passing reference to elections. With regard to the 2011 referendum on the electoral system and Brexit, it examines binary and multi-option voting; then, going a little more technical, it shows that the collective will can be identified with preferential voting.

Majority voting

Two-option questions may be necessary in law: guilty or not guilty? In politics, however, topics like the electoral system or our relationship with Europe need not be reduced to binary ballots: “option A or B?” or worse, “A, yes or no?”

Indeed, there may be only one political question where a binary choice summed up the entire debate: “Which side of the road shall we drive on?” Even then, in Sweden’s 1955 referendum, there were three options on the ballot paper: ‘left’, ‘right’ and ‘blank’. So those indifferent could also vote, choosing ‘blank’ and thus supporting the winning option; everyone was able to vote positively.

Basically, political questions should not be closed dichotomies: “Are you Protestant or Catholic? Serb or Croat? communist or capitalist? for or against?” Decision-making should not be win-or-lose, as facilitated by the most ancient, primitive, divisive and inaccurate measure of collective opinion ever invented.

Unfortunately, politicians like to control things. In 2011, David Cameron decided that a referendum would be “A v ”: first-past-the-post, (fptp), versus the alternative vote, (av). For supporters of proportional representation (pr), that’s like asking a vegetarian, “beef or lamb?” When New Zealand debated electoral reform, an independent commission drew up a short list of five options, from fptp to the Irish form of av, proportional representation, single transferable vote (STV), pr-stv. So (nearly) everyone could vote positively and New Zealand, like Germany, now has half fptp and half pr-list.

There were at least four options on Brexit: A, the EU; B the EEA; C the Customs Union and D, the WTO, but Mr Cameron decided this vote would be, in effect, “A, yes-or-no?” So B, C and D supporters voted ‘no’ – or ‘leave’. If the question had been “B, or C, or D, yes-or-no?” doubtless the answer would also have been a majority ‘no’. To state, then, that this ‘remain-or-leave’ ballot identified the will of the people is as illogical as to claim, on the basis of the 2011 referendum, that the British people prefer fptp to pr.  

Multi-option voting

If the choice is binary, a (simple or weighted) majority vote is the only appropriate methodology. But in politics, as noted, complex questions should not be reduced to two options. Moreover, no one individual should be able to dictate the referendum question, (as did Napoleon, Hitler et al).

If there are three or more options, several decision-making procedures are available. They include plurality voting; the two-round system, (trs); av; and a points system.

Plurality voting is like the UK’s fptp; voters choose just one option from a ‘plurality’ of them, and the option with the most votes wins.

With trs, the French electoral system, if nothing wins 50%+ in the first round plurality vote, a second round majority vote is held between the two leading options.

av is used in Australian elections, while pr-stv is popular in Ireland. In a series of plurality votes, the least popular option is eliminated and its votes transferred as per its voters’ wishes… until one option wins a majority.

The modified Borda count, (mbc), is a points system. With four options, a 1st preference may get 4 points, a 2nd 3, a 3rd 2, a 4th 1… and the option with the most points wins. The bc is used in Slovenian elections and, a slightly different version, in Nauru.

Consider, then, Table I, a voters’ profile in which 14 voters cast their preferences on options E, F, G and H. Option E is obviously very polarised, is also divisive, G is a bit better, but it seems H best represents the consensus.

Table I            A Voters’ Profile



Number of voters

























Plurality voting counts only the 1st preferences, so the social ranking is E 5, F 4, G 3, H 2, and the social choice is E.

In trs, assuming the voters’ preferences stay unchanged, the second E v F round gives E 5 F 9, so F tops this poll.

With av, H loses stage (i) and its 2 votes go to G. The stage (ii) count is E 5, F 4, G 5, so F’s 4 votes now go (not to H which has been eliminated but) to G so, stage (iii), it’s E 5 G 9, and G wins.

Finally, an mbc gives E (5 x 4 + 9 x 1 =) 29, F 31, G 36 and H 44, and the outcome is H.

So the democratic choice is E, or F, or G, or H. But, as suspected above, option H best represents the collective will. With this particular profile then, as with many others, the ‘alternative’ outcomes of E, F and G are ‘post-truth’ popularities, to quote current jargon. Likewise, the (almost binary) US presidential contest meant that Donald Trump’s victory was ‘fake’.

Preferential voting

Of the above methodologies, only the mbc counts all the preferences of all the voters. Little wonder, it is the most accurate, (as is the Condorcet criterion). At best, the mbc identifies the option with the highest average preference, and an average of course involves not just a majority but every voter; the mbc is win-win.

Moreover, as long as the options are finalised independently, it best qualifies under Kenneth Arrow’s ‘impossibility theorem’. Indeed, according to Sir Michael Dummett, it “is the soundest method of identifying the… most acceptable” option, while Don Saari concludes, “only [it] offers an accurate accounting of the voters’ preferences.”

Let’s take the argument further. If the Brexit options – (A) EU, (B) EEA, (C) Customs Union and (D) WTO – are listed in order ‘soft’ to ‘hard’, then the voters’ preferences may follow a pattern. If Mr i's 1st preference is D, he may have a 2nd of C, a 3rd of B and a 4th of A. Similarly, Ms j and Ms k, whose favourite is B, may have a 2nd preference of A or C, a 3rd of C or A, and a 4th of D

Table II          Single-peaked Preferences


Table II shows that these sets of preferences are all ‘single-peaked’. If Mr l, however, has preferences of 1st C, 2nd A, 3rd D and 4th B, his set, suitably dotted, will have two peaks, as in Table III.

Table III         An Oddity


If the voters do cast single-peaked preferences, the points totals will also be single-peaked. The peak of this collective set, as in Table IV, is the “will of the people”. Furthermore, the sheerer the summit, the greater the degree of consensus. It is an exact science.

i (1, 2, 3, 4) + j (3, 4, 2, 1) + k (2, 4, 3, 1) = i,j,k (6, 10, 8, 6)

Table IV         Vox Populi


Consensus decision making

In summary, to identify the will of the people:

  • Nothing complex and/or controversial shall be binary. Instead,
  • all relevant options shall be allowed ‘on the table’;
  • an independent authority shall draw up the final (short) list of options;
  • those concerned shall cast (one, some, or all) their preferences; and
  • if the option with the most points passes a pre-determined threshold, it shall be enacted.

The current scenario 

The outcome of the Brexit vote was at best unclear. Any future binary referendum like the Lib-Dem proposal would be equally vague and could lead to a never-end-’em. Similarly, in any binary parliamentary vote, “mps would have a choice between only [Mrs May’s] terms and no deal at all,” so, to quote Gus O’Donnell, (The Guardian, 21 February), it would “amount to nothing.”

The danger, of course, is that the world’s obsession with majority voting/rule will continue; that Le Pen, Wilders etc. will dictate their own referendum questions; and that the forces of populism will continue to vote ‘no’… to everything… until there is nothing.

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