Pulling the Goalie Makes Your Team Worse. That’s Exactly Why You Do It.
Down one, ninety seconds left. Pulling your goalie makes your team worse — you will be scored on more often than you score. It also lifts your chance of winning from about 3% to about 5%. That is not a paradox. The favorite and the underdog are not trying to maximize the same thing.
The question: when is it correct to deliberately make your team worse?
The answer. Down one with ninety seconds left, pulling your goalie lowers your expected goal margin; you will be scored on more often than you score. It also lifts your chance of winning from about 3% to about 5%.
Why that is surprising. We are trained to maximise the average. Here the right move makes the average worse, and it is still right.
Yesterday’s piece argued that a matchup is decided in the slots that swing, not the ones that score most; variance is where the outcome lives.
It ended with a warning: variance is leverage, and leverage cuts both ways. The boom-or-bust flex is where we win the week and where we lose it.
Today, the resolution. Whether you reach for volatility or run from it has nothing to do with your roster or your nerve; it depends on one thing, which is whether you walked in ahead or behind.
One line runs the whole thing
Write down what you actually want. Not points — points are the currency, not the goal. You want your total to land above theirs.
Say the projected margin between you is m, and the week-to-week swing in that margin is s. Your chance of winning comes down to the ratio: how far ahead you are, measured in units of how much things move.
Everything follows from that fraction. Grow the top — get better players — and you win more. But grow the bottom and the fraction shrinks toward zero whichever side we are on; a fraction of zero is a coin flip.
So the underdog should crank up the swing. Can he? Barely, and this is the part the strategy columns skip.
First the good news. Volatility is genuinely free. Among 2024 flex starters projected between six and nine points a week, the steadiest third swung about ±4.5 points. The swingiest third swung about ±7. Same projection, half again as much motion.
Jakobi Meyers and Travis Etienne were the metronomes. Amari Cooper and Tank Bigsby were the slot machines. If you are behind, take the slot machine. It costs nothing.
Now the bad news. One flex slot is about a seventh of your roster’s variance, and variance adds up as squares. So swapping the metronome for the slot machine moves the margin’s swing from about 25 points to about 25.5.
For a manager projected to lose by twelve, that is worth less than half a percentage point.
Build the whole lineup out of the swingiest third at every position and you do better. The swing goes from 23 to 28, and the underdog climbs from 30% to 34% while the favorite slides from 70% to 66%. Real, and directionally certain — and still only about four points.
The lever that actually works: take away the trials
Now notice the clue. If independence is what smothers the swing, then real volatility does not come from making each piece wilder; it comes from using fewer pieces.
Watch the two quantities. A talent edge grows with the number of chances. The noise around it grows only with the square root of that number. Cut the game in half and you cut the favorite’s edge in half while shrinking the noise by just thirty percent. The gap closes.
Play a game of one possession and we are back to a coin flip.
| Possessions each | Favorite’s projected margin | Swing in the margin | Underdog’s chance |
|---|---|---|---|
| 60 — track meet | −15 | 24 | 27% |
| 48 — normal game | −12 | 22 | 29% |
| 24 — grind it out | −6 | 15 | 35% |
| 12 — pure chaos | −3 | 11 | 39% |
Consider how much of sport this explains. The underdog runs the ball and lets the clock bleed. The overmatched team huddles while the favorite wants tempo. A seven-game series is kinder to the better team than one elimination game, and every March the one-seed dreads the team that walks the ball up the floor.
Notice that none of it is timidity. Shortening the game is the most aggressive thing an underdog can do, because it is the only lever big enough to move the bottom of that fraction.
And the purest version is the goalie.
Down one with ninety seconds left, keeping him in gives you a good defense and almost no offense. Your model chance of winning is about 3%.
Pull him and you roughly double your own scoring rate, while letting theirs run three or four times higher. Your expected goal margin over that ninety seconds falls by about a seventh of a goal. You have made your team measurably worse.
Your chance of winning goes up by two-thirds, to about 5%. Because the only outcome that was ever worth anything was the one where you score, and the empty net is the price of buying more chances to do it.
So variance is neither good nor bad; it is a drag toward the coin flip, which threatens us when we are ahead and rescues us when we are behind.
The ProfessorWhat to take home: ask what you are actually maximizing
Underneath all of it is a question people almost never ask before they start optimizing.
Expected points and probability of winning sound like the same goal; they are not, and we conflate them constantly. When the payoff is a threshold — ahead or behind, in or out, made the cut or did not — the average stops being the thing to chase. The right move can be one that makes your average worse.
So the favorite and the underdog face identical facts, run identical math, and correctly arrive at opposite strategies. Same fraction, read from opposite sides of zero.
Remember that past the arena, because the announcer never does.
When the trailing coach onside-kicks and it fails, it was reckless. When the leading coach punts and holds on, it was disciplined. Both verdicts are handed down afterward, from the result, about a decision we had to make before it.
“Conservative” and “aggressive” get talked about as personality types. They are positions. If you are ahead, smother the swing. If you are behind, go find some.
And if someone tells you a decision was foolish because of how it turned out, they are grading the ending, not the choice.
Notes & sources
Fantasy figures use 2024 regular-season weekly fantasy points (standard scoring) for genuine starters — at least twelve games — from nflverse weekly data, the same source as the companion piece. Win probabilities come from treating the margin between two lineups as normally distributed and reading off how many swings above zero the projection sits. The matched comparison is deliberate: across all flex-eligible starters, weekly average and weekly swing correlate at about 0.9, so the volatile players are usually just the better players. Comparing only within a narrow projection band (six to nine points) is what isolates volatility from talent — otherwise you would be measuring the wrong thing entirely.
The possession table and the goalie calculation are models, not observed results, and are labeled as such: a favorite worth a quarter-point per possession with a typical per-possession scoring spread, and a hockey team down one with ninety seconds left whose pulled goalie doubles its own scoring rate and quadruples its opponent’s. Both conclusions are robust to reasonable changes in those inputs — the direction is arithmetic, the exact percentages are not gospel.
Two primers sit underneath this one: Variance (Concept No. 23) for why swing decides outcomes at all, and The Objective Function (Concept No. 24) for the difference between maximizing an average and maximizing your chance of finishing ahead. The full archive is at thesportspage.net.
