Social Discounting Enables Fast and Reliable Collective Escape

Solitary animals face a tradeoff when detecting threats: faster detection means accepting more false alarms. We show that groups can manage this tradeoff better by treating an undisturbed neighbor as evidence against a threat, becoming both faster and more accurate than lone individuals. Modeling each animal as a noisy evidence-accumulator that flees when its belief crosses a threshold, we find that a neighbor's flight signals danger while its stillness signals safety. A naive responder reacts only to flights and inflates false alarms as the group grows; a Bayesian responder weighs both, approximated by a single social discounting rate that interpolates between these limits. This yields closed-form expressions for group performance, including cascade branching ratios that stay strongly subcritical in safety and turn supercritical under threat, so the rate at which an animal discounts a threat while its neighbors stay still can be inferred from behavior alone, and it sets a ceiling on how many neighbors an animal can attend before discounting alone can no longer hold its false-alarm rate. Wild sulphur molly shoals under bird attack are best described by discounting rates well above what individually Bayesian updating supplies over any neighborhood they could plausibly attend, and the same model, at the inferred value, predicts a false-alarm rate that stays constant as shoals grow.