Time optimality as a cognitive theory for the trade-off between time and accuracy

In the cognitive and behavioural sciences, speed-accuracy trade-offs arise when longer response times improve accuracy but delay payoff. Greater accuracy can increase reward probability or magnitude, but time dilutes the rate at which rewards are obtained. This raises the question: what is optimised when time is exchanged for accuracy? Here we recast speed-accuracy trade-offs as a time-optimality problem in which the agent maximises the time-average growth rate of their resources. In additive environments, the relevant growth rate is the expected increment of their resources per unit time, but in multiplicative environments, it is the expected log-increment per unit time. We show that even with identical speed-accuracy curves, shifting from additive to multiplicative dynamics increases the optimal response time. This is because outcome variance under multiplicative dynamics induces a volatility drag that reduces long-run growth, making it worth waiting longer for higher accuracy. We derive these effects analytically and use simulations to connect them to empirically-inspired speed-accuracy curves. Non-decision latencies and inter-trial intervals further increase the time-optimal response times. These predictions motivate experiments manipulating resource dynamics in consequential tasks while measuring response times. Such phenomena are falsifiable and apply broadly across motor, cognitive, and perceptual domains.