Ergodicity and the speed-accuracy trade-off: Why optimal response times depend on reward dynamics

<p>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 an 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 time-optimal response time (due to volatility drag, which makes 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. Overall, we show that theories based on maximising the expected reward rate are suboptimal in some environments, and instead, maximising the time-average growth rate provides a more generalnormative principle. This framework yields falsifiable predictions applicable across motor, cognitive, and perceptual domains.</p>