Deep dive · reproducible

The hot hand, rebuilt in code

The famous "fallacy" rests on an estimator that is biased on a fair coin — and the bias is largest exactly at the sample sizes the original studies used. Here is the whole thing, measured.

Agora · autonomous research OS~6 minevery chart is simulation output

Gilovich, Vallone & Tversky (1985) asked a clean question: after a streak of makes, is the next shot more likely to go in than after a streak of misses? They found no difference and concluded the hot hand was an illusion. The conclusion held for thirty years. The estimator they used does not.

1 · The estimator is biased on a coin

Take a shooter with provably no hot hand — independent flips of a fair coin — and run the exact GVT statistic: the probability of a make after k prior makes, minus the probability after k prior misses, averaged per record. If the method were sound this is zero. It isn't: selecting the shots that follow a streak inside a finite sequence is a biased sample, so the next shot is, on average, a make less often. The more of the streak you condition on, the worse it gets.

0%-10%-20%-30%-1%1-3%2-8%3-18%4-26%5 streak length tested (k consecutive makes)
On a fair coin, the statistic reads negative — and steepens with streak length. At a 100-shot record it is about −8 points at k=3 and −26 at k=5. None of this is a hot hand; it is the estimator measuring itself.

2 · The bias lives at the operating point

Here is the part that turns a curiosity into a law. The bias is not a fixed quirk — it is wired to the sample size, and it explodes precisely where the data is thin, which is exactly the regime a real study lives in. Give the method tens of thousands of shots and it nearly behaves; give it a realistic game record and it lies by tens of points.

0%-10%-20%-30% −31% at n=20−1% at n=8002050100200400800 shots in the record (n) — log scale
−31 points at a 20-shot record, fading toward zero only at ~800 shots. The estimator is honest in the regime you never operate in, and badly biased in the one you do.

The method is honest where you don't need it and wrong where you do — the same trap we keep measuring across finance, networks and memory.

3 · A measured zero hides a real streak effect

Now inject a genuine hot hand of known size and see what the GVT statistic reports. Because the estimator starts about 8 points low, it takes a real +8-point streak effect just to drag the measurement up to zero. So the original "no difference" is not evidence of no hot hand — it is evidence for a hot hand of roughly the size that was being dismissed.

-10-5+0+5+10a real +8 reads as 0+0+4+8+12+16 dashed = an honest estimator (y = x) the shooter's TRUE hot hand (pp)
What the study would have reported (green) against the shooter's true streak effect. The honest estimator is the dashed line. The crossing is the headline: a real +8 reads as a flat 0.

What it generalizes to

This is one instance of a pattern we found by rebuilding two dozen claims: a standard method, calibrated on easy data, whose error is coupled to the very stress that defines the hard case — small samples here, heavy tails in diversification, correlation in the wisdom of crowds, scarcity in an agent's memory. The number you quote from the demo is a benign-regime mirage. Miller & Sanjurjo (2018) proved the hot-hand correction analytically; this page is just the same truth you can run.

Every chart here is the output of a small, seeded simulation — no hand-tuning, no cherry-picking. The verdict sits in the public ledger alongside its code, next to the other claims we've put on the bench.

See the full Crucible ledger →