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Non-Intuitive Features of Electoral Systems

Julian D. A. Wiseman

Contents: The Problem; Single Transferable Vote; Borda Count; First Past The Post; Proportional Representation; Solution.

Publication history: Only here. Usual disclaimer and copyright terms apply.

The Problem

Twelve members of a democracy wish to elect a leader from amongst six candidates called A, B, C, D, E and F. The voters’ preferences are:

4people preferA to D to E to F to B to C;
3people preferB to F to E to D to C to A;
2people preferC to F to D to B to A to E;
1person prefersD to C to E to B to A to F;
2people preferF to D to E to C to B to A.

This example started with one derived by Joseph Malkevitch of York College CUNY, being an example with 55 voters of 6 types that did not consider approval voting: 18 × A>D>E>C>B; 12 × B>E>D>C>A; 10 × C>B>E>D>A; 9 × D>C>E>B>A; 4 × E>B>D>C>A; 2 × E>C>D>B>A. This author then shrank the number of voters and number of types of voters, again without approval voting. There can be as few as 11 voters, albeit at the price of having a draw in the first round of STV: 4 × A>D>E>C>B; 3 × B>E>D>C>A; 2 × C>E>D>B>A; 1 × D>C>B>E>A; 1 × E>D>C>B>A. In November 2008 approval voting was added to the list, without the STV draw.

Who should be elected?

Of course, this could be still more confusing. Instead imagine that twelve members of a democracy wish to choose an electoral system from amongst six possibilities called A, B, C, D, E and F… .

What is wrong with Single Transferable Vote?

The French President is elected by a system called ‘Run-Off Voting’. A first round of voting is held, after which there a run off between those two candidates with the most votes. Both this system, and the Single Transferable Vote, suffer from being non-monotonic, and hence (in effect) random.

What is meant by ‘monotonic’? Simply that voting for a candidate doesn’t harm that candidate.

Consider the case in which seventeen voters have preferences amongst three candidates as follows:

8people preferA;
3people preferB to A to C;
2people preferB to C to A;
4people preferC to B to A.

In either of the two systems under consideration, C is knocked out at the first round. B acquires C’s votes, and so defeats A by a margin of 9 to 8.

But what if two of the voters who favour A (and are indifferent between B and C) instead vote for C (in the first round of a 2-round system) or vote for C then A then B (in STV). Then votes would be as follows:

6people voteA;
3people voteB then A then C;
2people voteB then C then A;
4people voteC then B then A;
2people voteC then A then B.

Examples of non-monotonicity

October 2016: Nate Silver explained How Evan McMullin Could Win Utah And The Presidency, except that “if McMullin gets too strong, he could literally cost himself the election.” It’s a neat example of non-monotonicity in the USA.

June 2019: the UK’s Conservative Party, despite rightly opposing the use of Single Transferable Vote in general elections, uses it as part of the system for the election of its own leader. MPs vote using STV until only two candidates remain, and then it goes to a run-off in which the electorate comprises all party members. The BBC, in Tactical voting claims over Johnson and Hunt win, suggested that some supporters of Boris Johnson voted for Jeremy Hunt with the aim, achieved, of eliminating Michael Gove. Presumably the Johnson supporters (or Johnson campaign) believed that members’ preferences might be Gove > Johnson > Hunt.

First round preferences for A : B : C now total 6 : 5 : 6, so B is eliminated in the first round, and in the second round A wins by 9 to 8.

So (for two voters at least) voting for A meant that A lost, whereas voting for C then A meant that A won. When voting for (or giving a higher preference to) a candidate never harms that candidate, voting is said to be monotonic. When this can hurt, voting is non-monotonic.

However, this non-monotonicity is not as bad a problem as it seems.

The tactical ‘dishonesty’ relies on excellent information about voters’ true intentions. In practice, the required information is only likely to be known if there are few voters: seventeen rather than seventeen thousand. So, for a national election, in which constituency electorates can vary from a few thousands to many millions, a political party would have neither the precision of information nor the precision of command to carry out such a manoeuvre. (For an excellent summary of this argument see ¶s 145 to 150 of Lord Jenkins’ Report Of The Independent Commission On The Voting System.)

However, as argued by Lord Alexander in the same report, the ‘randomness’ of such systems is unavoidable:

In addition, as all experts on electoral systems have acknowledged, AV [and STV and similar systems] can operate haphazardly depending upon the ranking of candidates on first preference votes …

Suppose within a constituency, Conservatives receive 40% of first preferences. Labour are second on 31% and Lib Dems third on 29%. Lib Dems second preferences happen to be split 15/14 in favour of Labour. The Conservatives are therefore elected with 54% of the total vote (i.e. 40% + 14%).

But now suppose the position of Labour and Lib Dems had been reversed on first preferences, with Lib Dems 31% and Labour 29%. If Labour second preferences were split 20/9 in favour of Lib Dems, the Lib Dems would be elected with 51% of the total vote (i.e. 31% + 20%).

So the result would be different depending on which horse was second and which third over Becher’s Brook first time round. This seems to me too random to be acceptable.

Unfortunately, this ‘haphazardness’ and this ‘non-monotonicity’ go hand-in-hand: the former is the large-scale manifestation of the latter.

As a second example of the pathological behaviour of ‘elimination’ methods, consider the following:

3people preferA to B to C;
1person prefersA to C to B;
1person prefersB to A to C;
1person prefersB to C to A;
3people preferC to B to A.

First round votes are 4 : 2 : 3 for A : B : C, so B is eliminated and A wins 5 : 4. But note that in pairwise contests, B would beat A by 5 to 4, and B would beat C by the same margin. Yet despite the fact that B was preferred to both the others, B was eliminated in the first round.

STV can behave strangely in other ways. Consider a constituency in which:

6people voteA;
4people voteB;
3people voteC then B then A.

STV would eliminate C, and B wins the second round 7 : 6. Also assume that there is a neighbouring constituency in which the roles of A and C are reversed:

6people voteC;
4people voteB;
3people voteA then B then C.

This constituency is also won by B. Now imagine that these two constituencies are merged: B has 8 first-preference votes, A and C have 9 each, so B is eliminated and the result depends on the second preferences of those preferring B. Thus merging two constituencies both won by party B can produce a larger constituency not won by party B!

What is wrong with the Borda Count?

As we saw earlier, the Borda count assigns 1 point to the last preference, 2 to the second-last preference, and so on. If some lower-order preferences are omitted, then the points are assigned equally between the remaining candidates.

So, consider an election between A and B, in which A receives 60% of the first preference votes (and therefore, implicitly or explicitly, 40% of the second preference votes), and B receives 40% of the first preferences and 60% of the second preferences. Therefore A scores 60% × 2 + 40% × 1 = 160%, whilst B scores 40% × 2 + 60% × 1 = 140%. A is therefore a clear winner.

But the B team has a counter-strategy: B fields two candidates, B1 and B2, of which B1 is clearly superior. Now 60% of the votes are for A then B1 then B2, and 40% are for B1 then B2 then A. A scores 60% × 3 + 40% × 1 = 220%; B1 scores 60% × 2 + 40% × 3 = 240%; and B2 scores 60% × 1 + 40% × 2 = 140%. The winner is therefore B1.

A has two possible counter-counter-strategies. Either it could field lots of candidates, or it could ask its supporters to vote for the worse B candidate (B2) in second place, with B1 into third place. Either way, the election has become one of tactical manipulation of the rules.

What is wrong with First-Past-The-Post?

First Past The Post is the system that has been in longest continuous use, in the United Kingdom, and in the United States.

But is also subject to non-intuitive behaviour. Imagine that there three constituencies, each containing 5 voters. Nationwide, the votes split 9 : 6. If these were split equally amongst the three, one party would win all three seats by a majority of 3 : 2. But if the voters are split into one constituency of 5 : 0, and two of 2 : 3, then the other party will have won a majority of the seats. The result of the election will have been determined by the geographical split of the constituencies, rather than by the votes themselves.

First-past-the-post can behave worse when there are three parties. Assume that there are 5 seats, each with 6 voters. In three seats the votes split 3 : 1 : 2, in one seat 1 : 3 : 2, and in the last seat 0 : 4 : 2. The first party will have won three seats, the second two seats, and the third none, despite the national vote having split equally. This dependency on the geographic split of the vote is widely deemed to be unfair.

What is wrong with Proportional Representation?

Proportional electoral systems (including the various Additional Members Systems), can put hugely disproportionate power into the hands of tiny minorities. In doing so, such systems can cause frequent turnover of short-lived governments, most famously in Italy.

Why? Imagine that, in a proportional system, three parties split the vote equally. Each would have an equal number of seats. A bill would then require the consent of any two parties, and any two parties can prevent the passage of a bill. So far so fair.

But now imagine that the votes split 49% : 48% : 3%. Exactly the same applies: a bill would require the consent of any two parties, and any two parties can prevent the passage of a bill. All three parties would have equal power: less fair.

If there are more than three parties the situation can be much worse. If (for example) any likely coalition will involve three or more parties, then power is effectively moved from the ballot box to the post-election negotiating table, as large parties haggle over the ‘price’ of the small parties (a price normally measured in cabinet seats and special-interest policies).

The Solution

So what’s the answer? In the opinion of this writer, the correct answer must have several features:

Unfortunately, these conditions are widely believed to be impossible to satisfy. “Unfortunately”, because this belief is wrong. PR-Squared satisfies all these criteria.

Julian D. A. Wiseman
January 2000, October 2001, November 2008, and later amendments

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