Contents
The Square Law of Strength comes from military theory. Its basic idea can be applied to any system in which the winner of each individual contest takes everything and the loser gets nothing. Britain’s first-past-the-post electoral system is such a system. On May 7, 2015, it showed how powerfully concentrated strength works, and how little widely scattered strength achieves.
Two parties, two calculations
Situation
The Conservatives win 330 of the 650 seats with 36.9% of the vote, and with them an absolute majority. Labour gets 232 seats with 30.4%. Behind them are two parties with opposite results. The SNP receives 1,454,436 votes – 4.7% – and 56 seats. UKIP receives around 3.9 million votes – 12.6% – and one seat. The Liberal Democrats get eight seats with 7.9%, and the Greens one with 3.8%.
Link to the principle
On paper, the SNP needed around 26,000 votes per seat. UKIP needed almost 3.9 million for its one seat. It had more than two and a half times as many votes as the SNP and got one fifty-sixth of its seats. The payoff follows not quantity but concentration.
How first-past-the-post counts
Situation
In each of the 650 constituencies, the candidate with the most votes wins the seat. Votes for those in second and third place bring no seats. There is no compensation through party lists.
Link to the principle
This follows the same logic that Frederick William Lanchester described for combat: whoever is superior in an individual contest takes everything; whoever is inferior loses everything. Strength that is present a little everywhere fizzles out. Strength that crosses the threshold in one place becomes fully effective. The payoff therefore does not grow evenly with the votes; it jumps as soon as the concentration is sufficient.
The SNP: everything in one country
Situation
The SNP stands only in Scotland, in the 59 Scottish constituencies. After the 2014 independence referendum, which the supporters of independence lost, the party grows strongly. Under Nicola Sturgeon, First Minister of Scotland from November 2014 to March 2023, it wins around half of the vote in Scotland in 2015 and 56 of the 59 constituencies.
Link to the principle
The SNP does not have to spread its strength. Its entire electorate lives in one area. There, it is the strongest party in almost every constituency. Measured against the whole United Kingdom, it is small; at the place where the outcome is decided, it is superior.
UKIP: a little everywhere
Situation
UKIP under Nigel Farage stands across the whole country. It wins sizable shares in many constituencies but comes first almost nowhere. Its only seat is won by Douglas Carswell in Clacton in the county of Essex. Farage himself is defeated in the constituency of South Thanet.
Link to the principle
The party has a lot of strength, but it is spread out. In almost every single constituency, it goes in outnumbered. By the logic of the principle, that is the worst possible position: high effort, hardly any return.
Limits of the comparison
Situation
Elections are not battles. The SNP did not systematically gather its voters in Scotland; it is a Scottish party. UKIP could not simply have redistributed its votes. The first-past-the-post system itself is politically contested, and critics see the 2015 result as proof of its distortions.
Link to the principle
The example does not demonstrate Lanchester’s formula but its basic idea: wherever the locally stronger side wins everything, the distribution of strength matters more than its quantity. That also applies to sales, advertising or church work. If you are a little present everywhere, you are not strong enough anywhere.
Key takeaways
Key lessons
- Ask whether, in your field, the locally strongest takes everything.
- If so, the distribution of your strength matters more than its quantity.
- Cross the threshold in a few places instead of narrowly missing it everywhere.
- Measure success by the places you win, not by your overall share.
- Check whether your strength is chosen or just an accident of your situation.
Summary
In the 2015 general election, UKIP got more than two and a half times as many votes as the SNP, but 55 fewer seats. The SNP was concentrated in Scotland and was the strongest force almost everywhere there. UKIP was present everywhere and ahead almost nowhere. First-past-the-post rewards concentration disproportionately and punishes dispersion just as much. That is exactly the basic idea of the square law.
Sources
- Frederick William Lanchester: Aircraft in Warfare: The Dawn of the Fourth Arm. London: Constable 1916.
