A lone fish sees a shadow overhead. Should it flee? If it's just a leaf, it wastes energy and gives away its position to a predator. If it's a heron, hesitation is fatal. That's the eternal trade-off of the solitary individual: quick reactions mean many false alarms, while caution means risking being eaten.

In a school, it's different. A neighbor that calmly keeps nibbling on algae is living proof that there's no danger. Scientists built a model where each fish accumulates ambiguous signals like drops in a glass and darts off when the level overflows. It turns out that a neighbor's flight is strong evidence of a threat, but its stillness is evidence against one. A naive fish reacts only to others' flights and, as the school grows, starts darting at every little thing. A smart fish weighs both: it's as if it discounts the threat while all is quiet around it.

This discount has a precise measure. From the school's behavior, one can calculate how much an animal devalues an alarm signal when neighbors are calm. And this number has a limit: if a fish watches too many neighbors, a single discount is no longer enough to suppress false alarms. Observations of wild mollies attacked by birds showed that their discount is much larger than what simple individual accumulation of signals could produce. That's why false alarms in the school don't increase even as the number of fish grows.

Thus, the school turns the solitary individual's weakness, inevitable errors, into strength. A neighbor's calm becomes as important a signal as its flight.