Coalition formation

Why can the party that won the most votes end up outside the government it expected to lead?

On election night a party takes more seats than any other, its leader gives a victory speech, and six weeks later somebody else is sworn in as head of government. Nothing improper happened. No recount changed anything. The arithmetic that produced the victory speech simply was not the arithmetic that decides who governs.

So which arithmetic does decide it — and why does coming first buy so much less than it appears to?

Start with what a parliamentary government actually needs. Not the most seats: the confidence of a majority. A cabinet stays in office only while more than half the chamber declines to vote it out. That is a threshold, not a ranking, and thresholds and rankings behave very differently.

Watch what happens when you take that seriously. Suppose a hundred-seat chamber where 51 votes carry a confidence motion, and four parties hold 45, 25, 20 and 10 seats. Ask not "who is biggest" but "which groups clear 51". A alone falls nine short. A with any single partner clears it: A+B is 70, A+C is 65, A+D is 55. But look at the trio without A: B+C+D is 55. That also clears it.

Now the question worth asking. In how many of those winning groups is a given party necessary — meaning that if it walked out, the group would drop below 51? Count them. There are eight winning combinations in all. A is necessary in six of them: in each of its three pairs, and in each of the three trios that contain it, since removing A from ABC leaves 45, from ABD leaves 35, from ACD leaves 30. In the four-party group nobody is necessary, because any one can leave and the rest still hold at least 55. And in B+C+D, all three are necessary — remove any one and the remainder is 45, 35 or 30.

So the tally of "times a party is the difference between winning and losing" runs A: 6, B: 2, C: 2, D: 2. Twelve in total. As shares: A holds 50 per cent of the leverage, and B, C and D hold about 16.7 per cent each.

Sit with that. B has 25 seats and D has 10 — two and a half times the seats — and in this chamber they are exactly interchangeable. Every group that needs B needs it in precisely the same way it would need D. The folk account, that seats convert into influence roughly in proportion, is not slightly off here; it is flat wrong across a fifteen-seat range.

And A, with 45 per cent of the chamber, holds no veto at all. B+C+D can govern without it. That is the whole answer in one line: the largest party is excluded not by unfairness but because there exists a majority that does not contain it.

Why would the others take that option? Because seats are not the only currency. If A sits at one end of the main political dimension while B, C and D are strung along the other, the trio is closer together than any pair involving A. Add the commonly observed refusal — parties declaring in advance that they will not serve with a particular party — and A can be arithmetically ideal and politically untouchable at once.

Three refinements hold the reasoning together.

First, the numbers above are an illustrative chamber I chose to make the effect visible; nothing about 45/25/20/10 is empirical. What is not illustrative is the counting method, which follows deductively once you accept that governing requires a majority.

Second, the count of "times a party is necessary" is a formal power index, and it measures one thing only: bargaining leverage arising from arithmetic. It is silent about who becomes prime minister, which often turns on who is invited to try first — and that invitation, in most systems, does go to the largest party, which is exactly why its exclusion feels like a violation.

Third, and most awkwardly, this account makes a prediction that the evidence does not fully support. If leverage were pivotality, coalition partners should divide cabinet posts in proportion to their pivotality — so B, C and D should take equal shares. The recurring empirical finding, usually named after Gamson, is that portfolios are divided close to proportionally to seats among the parties in government (a recalled result, though a well-replicated one). So pivotality appears to govern who gets in, while something much closer to raw seat share governs what they divide once inside. My account explains the first and not the second; what covers the gap is that once a coalition exists, the relevant threat is no longer "form a different majority" but "walk out and collapse this one", and a party's capacity to make that threat credible tracks its size more nearly than its pivotality.

I have deliberately quoted no real election's seat totals. The threshold varies with chamber size, abstentions, and whether a country requires a positive investiture vote or merely the absence of a defeat — so any real figure would be doing convention-dependent work under cover of looking factual.

• Why minority governments — which clear no majority at all — are stable for years in some parliaments and unthinkable in others.
• The complement already in this collection is Voting rules, which asks how ballots become seats and flags proportional systems as the place where the question shifts from "who wins" to "who is represented". This piece picks up exactly there: after the seats exist, and the winning is no longer the point.

Every term the collection defines is gathered in the glossary.