Elections using single seats with a plurality voting system tend to be very unfair on small parties. Even large parties that come second place will be heavily under represented and often the largest party with only a moderate lead in percentage votes will win by a landslide in the number of seats.

But exactly how unfair? In 1909 James Parker Smith while giving evidence to the Royal commission on Systems of Elections used the analogy of elections being like taking shovelfuls of marbles from a box of red and blue marbles. If the marbles in the box had 11000 blue marbles and 9000 red marbles, Smith claimed, the proportion of shovelfuls that had a majority of blue or red marbles would be in a proportion of the cube of 11 to the cube of 9. Smith claimed that this could be derived mathematically.

It is usually assumed that Smith’s marbles were voters and the shovelfuls were constituencies. But this would be like taking multiple opinion polls in which the sample size was the average number of voters in a constituency. In 1906 that was just short of 8000 which were it a random sample is large sample size. If that was really what Smith had in mind the advantage to the leading party would be far greater than a cube rule and a the lead of 5% votes that Smith suggested in his example could give an almost clean sweep in seats.

Graham Upton (Blocks of Voters and the Cube 'Law', British Journal of Political Science, Jul., 1985, pp. 388-398) basing his proposal on the work of L S Penrose suggested that if Smith’s analogy was qualified by assuming that the marbles were sticky so the shovelful of marbles would not scoop up a completely random bunch of marbles but quite small number of clumps of marbles. This is justified by the real world reality that constituencies include distinct housing areas of council estates (ie social housing) or commuter estates that have distinct voting patterns. He then went onto to derive from that assumption a mathematical model that could be reduced to the cube rule.

It does seem to me, however, that something like this may well have been what Smith intended. In his submission he nowhere says that each marble represents a single voter and arguably his submission implies that each marble represents a quite large block of voters. Smith stated that he had had help from his friend the mathematician Major MacMahon. (The Law of the Cubic Proportion in Election Results, M. G. Kendall and A. Stuart The British Journal of Sociology, Sep., 1950, pp. 183-196). As MacMahon does not seem to have published anything on the matter it is plausible that Smith simply dropped in on MacMahon and MacMahon simply produced something written on the back of an envelope on the spot. Any reconstruction of what MacMahon gave to Smith can only be a guess but Smith’s submission does a least give some pointers in that direction. Smith’s example is of 11,000 blue marbles and 9,000 red marbles. At the time the house of commons had 670 members. If those 20,000 marbles represented the entire British electorate of the time then by dividing 20,000 by 670 and rounding down we get 29. If we take a binomial distribution with 29 tries then the probability of getting 14 bunches of red marbles is 0.431 and of getting 15 is 0.293. The mean of those is 0.362. Therefore 36.2% of constituencies will have more than half red voters if those assumptions are correct. The cubes of 0.45 and 0.55 are 0.167 and 0.091 which would points to 35.4% of constituencies having a red majority which is pretty close.

I have, of course, no way of knowing whether this is actually how MacMahon came up with a mathematical justification for the cube rule. It is also actually is a binomial rule but a cube rule is a good approximation of that and a good deal easier to calculate. What it shares with all attempts to derive the cube rule is that is not a proof and rests on assumptions about how voters group together that are simplifications.

Using the cube rule then can only be justified if it fits the data. Looking at past elections throws up a lot of examples that fit the cube rule but also a lot don’t. Smith himself presented in his submission seven elections which fitted the cube rule reasonably well.(M. G. Kendall and A. Stuart, 1950, p185) Generally it is also taken apply only to two party systems though Terence H. Qualter has extended it to Canada which has long had a multi-party system.(Seats and Votes: An Application of the Cube Law to the Canadian Electoral System Canadian Journal of Political Science / Revue canadienne de science politique, Sep., 1968, pp. 336-344). Qualter did, however have to break Canadian elections down to subsystems by grouping together constituencies which the contestants were the same parties and then applying the cube rule to those subsystems.

Germany has a mixed system in which half the members are elected from single winner constituencies and the result is made proportional by allocating compensatory seats to those parties that have gained less than their fair share from the constituencies. Because the number of members elected from constituencies has far less effect on the final total won by parties than a plurality election, the fear of wasting ones vote is likely to have played a far smaller role in voters’ choices.

2017 German Elections, Constituencies | ||||
---|---|---|---|---|

| | Seats | by Cube Rule | Proportional |

CDU+ CSU | 17286238 | 231 | 222 | 115 |

SPD | 11429231 | 59 | 64 | 76 |

AfD | 5317499 | 3 | 6 | 35 |

DIE LINKE | 3966637 | 5 | 3 | 26 |

GRÜNE | 3717922 | 1 | 2 | 25 |

FDP | 3249238 | 0 | 1 | 22 |

2013 German Elections, Constituencies | ||||
---|---|---|---|---|

| | Seats | by Cube Rule | Proportional |

CDU+ CSU | 19777721 | 236 | 233 | 143 |

SPD | 12843458 | 58 | 64 | 93 |

DIE LINKE | 3585178 | 4 | 1 | 26 |

GRÜNE | 3180299 | 1 | 1 | 23 |

FDP | 1028645 | 0 | 0 | 7 |

AfD | 810915 | 0 | 0 | 6 |

Comparing the number of seats predicted by the cube rule with actual number of seats, the fit for the two main parties is very good. The difference between the actual number and the cube rule predicted seats for the smaller parties never exceeds three though this does of course a much greater proportional divergence. It is worth keeping in mind that though Die Linke does a bit better than predicted by the cube rule, this far less than a pure proportional result. It is hard to imagine that Die Linke would stick with that party if the German voting system as a whole was a plurality system.

Because Northern Ireland, Scotland and Wales have parties that do not contest seats on a UK wide level, applying the cube rule to the votes UK wide would be misleading. For English constituencies all seats (other than that of the Speaker) were contested by the Conservative, Labour and Liberal Democrat parties and other parties take sufficiently small percentage of the vote that their failure to contest all seats does not distort things too much..

2015 UK Election, English Results | ||||
---|---|---|---|---|

| | Seats | by Cube Rule | Proportional |

Con | 10448644 | 318 | 351 | 220 |

Lab | 8087653 | 206 | 163 | 170 |

Lib-Dem | 3599692 | 6 | 14 | 76 |

UKIP | 2098458 | 1 | 3 | 44 |

Green | 1065859 | 1 | 0 | 22 |

2010 UK Election, English Results | ||||
---|---|---|---|---|

| | Seats | by Cube Rule | Proportional |

Con | 9908019 | 297 | 335 | 214 |

Lab | 7039387 | 191 | 120 | 152 |

Lib-Dem | 6075613 | 43 | 77 | 131 |

UKIP | 858163 | - | 0 | 19 |

BNP | 531343 | - | 0 | 11 |

Green | 258950 | 1 | 0 | 6 |

By contrast from the German constituencies, the two English results are a poor fit even for the two main parties. The runner up party, (in both cases shown, the Labour party) did significantly better than the cube rule would dictate. Even though the cube rule originated from the experience of British elections it has been diverging for some time. Way back in 1979, Markku Laakso was suggesting that for Britain the cube rule needed to be reduced to a lower power (Should a Two-And-A-Half Law Replace the Cube Law in British Elections? British Journal of Political Science, Jul., 1979, pp. 355-362). On top of that boundary changes are overdue and it has tended to be Labour seats that seen falls in the number of voters.

But while Labour did better than predicted by the cube law the Liberal Democrats did even worse than the cube rule would predict. The 2010 result is especially striking in that the Liberal Democrats were close to moving up into second place. The Greens did manage to win a seat despite that not being predicted by the cube law. This was the result of heavy targeting of the seat (Brighton Pavilion) over a number of elections. Further successes in local government elections in the area of constituency had enabled the party to convince voters that voting for them would not be a wasted vote. However, even leaving aside that that one seat was five less than what a purely proportional distribution would give, the Greens were essentially copying tactics pioneered by the Liberal Democrats. It seems that such tactics are harder to put into effect when a small party has made gains and now has something to defend.

If anything then, the cube rule understates the mountain a third party has to climb in a plurality based voting system. But if that sounds like an argument against third parties remember it is also hard to see a move towards proportional voting without significant inroads by third parties.

But while Labour did better than predicted by the cube law the Liberal Democrats did even worse than the cube rule would predict. The 2010 result is especially striking in that the Liberal Democrats were close to moving up into second place. The Greens did manage to win a seat despite that not being predicted by the cube law. This was the result of heavy targeting of the seat (Brighton Pavilion) over a number of elections. Further successes in local government elections in the area of constituency had enabled the party to convince voters that voting for them would not be a wasted vote. However, even leaving aside that that one seat was five less than what a purely proportional distribution would give, the Greens were essentially copying tactics pioneered by the Liberal Democrats. It seems that such tactics are harder to put into effect when a small party has made gains and now has something to defend.

If anything then, the cube rule understates the mountain a third party has to climb in a plurality based voting system. But if that sounds like an argument against third parties remember it is also hard to see a move towards proportional voting without significant inroads by third parties.

Despite its faults the cube rule is a valuable first approximation. Even when it gets results wrong its predictions are floating around the results actually seen. While past elections of representative body are the best guide to how elections are likely to go, the cube rule can quickly produce a result from expected voting levels even in novel situations such as a split in a major party.