One captain or two? The mathematics of FPL captaincy

The captain scores double. Counted carefully, the armband adds one extra copy of the captain’s points; his first copy is already in your XI total. So a squad’s expected points in a gameweek are

and the only thing a captaincy choice changes is the second term. If Haaland is projected for 7 points, captaining him is worth 7 extra points, not 14. Counting 14 double-counts the 7 he scores anyway.

Own one captain, A, and you captain him every week. Own a second, B, and each week you captain whichever is projected higher. Over gameweeks t, the extra captain points from owning B are

Two things follow. The gain can never be negative, so a second option never hurts your captaincy. And B only earns anything in weeks he is projected above A, and then only by the margin. A player who is slightly worse than A every week adds nothing, however good he is.

With three or more options the formula is the same, with the maximum taken over all of them. Each extra option only counts in weeks it beats every other option you already own, so the gains shrink quickly.

Because only the weeks where B is ahead count, the best partner is not the player with the second-highest ten-week total. It is the one whose good weeks fall in A’s weak ones. Over gameweeks 6 to 15, Bruno is projected for 56.8 points and Calvert-Lewin for 48.8. Yet beside Haaland, Calvert-Lewin adds 1.4 captain points and Bruno 1.1: Calvert-Lewin’s one big week lands exactly where Haaland is weakest.

The official rules: the captain scores double. If he doesn’t feature at all in his match, the vice-captain scores double instead. If neither plays, nobody does. A one-minute substitute appearance counts as featuring: the captain keeps the armband and his 1 point becomes 2, and the vice-captain gets nothing extra.

If the captain appears with probability p and the vice-captain with probability q and expected points V, the vice-captain adds

An illustration, not a projection: suppose a premium captain has a 5% chance of missing a match without warning before the deadline. A vice-captain who plays 97% of the time and is projected for 6.5 adds 0.05 × 0.97 × 6.5 ≈ 0.3 points a week. But you always have some vice-captain. Paying for a better one only buys the difference: a vice projected for 6.5 instead of 5.5 is worth 0.05 × 0.97 × 1.0 ≈ 0.05 a week, about half a point over ten weeks.

Insurance pays only when the captain doesn’t play. Rotation pays when he plays but someone else is projected higher. They are separate, and the second is usually much larger.

Every pound on one player is a pound not spent on another, so the right question is never “is this player good?” but “does this squad score more than the best squad without him?” That is what an optimiser answers, and why ranking players one at a time goes wrong.

Points per £m is a useful first filter but not the answer. Over these ten gameweeks Haaland is projected at 4.5 points per £m, worse than Bruno (4.8) and far worse than Tavernier (9.4). Yet he is in the best squad, and leaving him out costs 16.1 points. Only eleven players score each week, so the points a slot produces matter more than the points per pound.

The useful price is the marginal one: how many points an extra £1m buys once the squad is already as good as it can be. Re-solving the best squad at budgets from £95m to £105m puts that at about 2.0 points per £1m over ten gameweeks. That is the yardstick for any premium: his captaincy gain has to be weighed against what the same money buys elsewhere.

Haaland is the most-owned player and the highest projected. Saka is £6.0m cheaper and only 3.2 points behind over the ten weeks. Bruno and Palmer are the other premiums managers most often pair with him; Palmer is flagged 75% fit, and the model has his start chance recovering from 0.70 this week to above 0.90 by GW9.

With Saka, the armband splits five weeks each. Bruno takes it twice and Palmer once. The optimal squad (next section) ends up with four options. Calvert-Lewin and Tavernier, both about £6m and bought for their own points, take one week each.

The captain pairs above ignore the rest of the squad. The real test is to build the best possible £100m squad with and without each premium and compare the totals. The optimiser, left free, picks Haaland and Saka, scoring 633.8 projected points over the ten gameweeks (558.7 from the XI and 75.1 from captaincy).

Forcing Bruno in costs 0.9 points, and none of it is captaincy. His two good weeks (GW12 and GW14) are exactly the weeks Calvert-Lewin and Tavernier already cover, so in a real squad he never gets the armband. Palmer is the same story at 1.7. Without Haaland, the optimiser buys Saka, Bruno and Isak and still finishes 16.1 behind.

As a captaincy option, barely. Adding Bruno to Haaland and Saka adds 0.4 captain points over ten weeks on paper, and nothing in the optimal squad, where cheaper players already cover his weeks.

But the budget curve shows when a third premium does make sense. With £102.5m or more, the optimiser buys Bruno as well, for his own points. A third premium follows from a bigger budget, such as higher team value; it isn’t a reason to stretch one.

Transfers add about 8 points over the ten weeks (9 free transfers, no hits), and none of it is captaincy: the captain points (75.5) and who gets the armband are the same as in the fixed squad. The order doesn’t change either: Haaland with Saka is still best, forcing Bruno in still costs a little, and going without a £9m+ partner still costs most.

One thing the solver does that you shouldn’t copy: left free, it sells Haaland for the Arsenal away week and buys him back. That only pays because prices are held fixed here; in FPL you keep only half of any price rise when you sell, and he may rise while you are out.