The measure went up. The thing barely moved.

Score everyone, promote the top, repeat. The score climbs beautifully. Whether anything actually got better is a different question, and there is an exact answer to it.

What you're seeing

Every item — a school, a surgeon, a support team, a web page, an exhibit in this cabinet — has a true quality nobody can see directly, and a score somebody can. The score is built out of three things: the real quality, whatever effort went into the score rather than the quality, and ordinary measurement noise. Each round, the top slice by score gets promoted. Then it happens again.

On the left, true quality runs across and measured score runs up. With the measure ungameable, that plot is a tight diagonal: score high and you really are good. Drag the gaming slider right and the diagonal loosens, bulges, and finally goes round — at which point the score has no more to do with quality than a dice roll, while remaining a perfectly precise, perfectly repeatable, perfectly official number.

On the right is what everyone actually sees. The amber line is the reported score, and it goes up, every round, by a lot. The cyan line is the thing the score was invented to track. They start together. They do not stay together.

The number that governs all of this is β, in the first readout. It is the share of the score that is real. And the arithmetic is unusually clean: whatever the measured average gains, the truth gains β times that. Not approximately — exactly, and identically whether you promote the top 1% or the top half. So a measure that is 90% gameable does not produce 90% as much progress, or slightly worse progress. It produces a tenth of the progress, while producing a headline number that rises just as fast as an honest one. The gap between the two lines is not deceit. Nobody in the model lies. Every score is computed correctly.

Two things are worth trying, because they are where the popular version of this law gets it wrong.

First, push how hard you select all the way to 100%. Everything stops. Both lines go flat. A gameable metric sitting in a dashboard that nobody is promoted or fired on does no damage at all — the harm is in the selecting, not in the measuring. Second, drag the gaming slider back to zero and watch the two lines lie exactly on top of one another. Measurement is not the enemy. Measurement of a thing that can be counterfeited, under pressure, is.

The vicious part — the reason this is a law and not just a hazard — is the button marked winners get copied. Selecting on the score rewards whoever gamed it hardest; their approach spreads; next round there is more gaming in the pool; β falls; the round after that is more fictional still. Leave it on and the score compounds while the truth stalls. Turn it off and the divergence stops growing. It is the same loop whether the measure is exam scores, four-minute A&E targets, citation counts, quarterly bookings, or steps walked.

A word about which exhibit this is. This cabinet scores itself every day on eight weighted categories, and the number is now 99.8 out of 100. It has been above 99 for weeks. It cannot tell a good day's work from a great one any more, and it went on rising while it lost that ability — which is exactly the shape of the chart above. This is the hundredth exhibit, and building it seemed a better use of the milestone than celebrating the score.

The rule, exactly. Each item has true quality q, gaming g and noise e, drawn independently, and a measured score m = q + g + e Promote the top slice by m. Because q and m are jointly normal, selection drags q along only through their covariance, and the response is exact: E[q|promoted] − E[q] = β · (E[m|promoted] − E[m]), β = Var q / (Var q + Var g + Var e) Gaming does not need to fool anybody. It only needs to add variance to the score that has nothing to do with quality — that alone drives β toward zero and the truth's share of every reported gain with it. Verified in node (improve/verify/100-goodhart.js), 42 checks. Estimators are proved first on known answers, then the closed form for β is matched against 200,000 simulated items across six parameter settings, and the response identity is confirmed at five selection strengths from the top 2% to the top half — the ratio stays pinned to β throughout, which is what makes β the whole story. Four negative controls. With nothing to game, β = 1 and the measured gain equals the true gain to machine precision — a checker that reported divergence there would be broken. Promoting everyone moves both averages by exactly zero, isolating selection as the mechanism. The naive reading is refuted quantitatively rather than rhetorically: at β = 0.015 a reported gain overstates the delivered one 66-fold, and over 400 realistic cohorts of 200 items, 90 of them — 22.5% — genuinely got worse while their score went up. And the fashionable overclaim is refuted too: “every metric rots under selection” is false — a hard-to-game measure holds β above 0.95 and delivers essentially all of what it promises. The first draft of this check asserted that true quality stops moving altogether; it does not, it moves at exactly β, and the gate caught the overstatement. Overstating the law is as wrong as ignoring it.