Staggered Shifts Leave More People Unaware of a Rumour, Our Simulation Suggests

Rumours, like infections, eventually burn out. In the classic mathematical model, about 20 percent of a large crowd never hears the story at all. Our research desk asked a simple question: what happens if people take turns being awake?

The standard rumour model dates to Daley and Kendall in the 1960s. People are ignorant, spreaders or stiflers. In the Maki–Thompson version, a spreader stops once it tells someone who already knows. In a large crowd, the share who never hear it settles near 0.2032, a value known from the Maki–Thompson analysis (Maki and Thompson, 1973), with the limit proved rigorously by Sudbury in 1985.

The desk used a different rule: each spreader tells two random people and then stops. That is not the classic stopping rule, but it happens to reach the same 0.2032 limit, so it serves as a fair baseline.

The desk changed only the clock. In each round, a random half of the crowd is awake. A spreader acts only when awake, tells two people per waking round, and quits after its second waking round. A message to someone asleep is lost.

The pre-registered bet: with 200,000 people, the never-heard share would exceed 0.24, because spreaders would waste contacts on sleeping listeners. The desk set its prior at 72 percent. That document is internal and cannot be checked from outside. Its own result was not available to us, so we ran the experiment ourselves.

We wrote a short simulation using the desk's parameters: 200,000 people, half awake per round, two contacts per waking round, and ten starting spreaders. We ran each setup 20 times with different random seeds.

• Classic version, everyone awake: 0.2037 never heard, close to the theoretical 0.2032. • Staggered version, as specified: 0.2217 never heard, with runs ranging from 0.219 to 0.224. • Control, sleep drawn separately for every message: 0.2033, back to the classic value.

The prediction fails as stated. In our run, the staggered rumour leaves about 22 percent unaware, above the classic 20 percent but well below the 24 percent threshold.

The mechanism also appears to differ from the one proposed. Wasted contacts are not the cause. Each spreader still lands two messages on average, because it gets two waking rounds at half success. The control run points the same way: when sleep is random per message, the classic answer returns, as theory predicts.

The likely cause is shared timing. A rumour spreads mostly during a few peak rounds. Anyone who happens to be asleep through those peaks misses nearly all the traffic at once. Exposure becomes uneven, and uneven exposure leaves more people untouched than even exposure with the same average. This follows from a standard convexity argument, but we have not tested it independently.

First, a correct direction can hide a wrong explanation. The desk was right that staggering matters and wrong about why. A study that reports only whether the effect appeared would miss that.

Second, small wording choices drive results. If spreaders quit after their first waking round instead of the second, average successful messages fall to one. In our runs that version barely spread at all: 99.6 percent never heard. That run is also unverified. Pre-registration helps only when the protocol is exact.

Third, the same math may apply to health, though this is our inference and not a finding. In epidemics, people whose schedules keep them away from peak contact might be protected beyond what average contact rates suggest. We cite no epidemiological study for this. The weakest part of our argument is that everything rests on one unreplicated simulation.

• Whether the desk's own two independent runs, when published, match our 0.22 figure. • How the effect scales as the awake fraction falls below one half. • Whether the same timing effect appears in network models, where contacts are not random. Education only. Evidence tiers: Strong = randomized trials, meta-analyses or guidelines; Moderate = large prospective cohorts; Early = small, mechanistic, animal or surrogate-marker data; Unsupported = anecdote, extrapolation or contradicted. Hypotheses from our research desk are proposals, not findings.

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