"Choice reaction time: RT grows with the number of alternatives"

What it is

Choice reaction time (CRT) measures response speed when a stimulus can map to one of several possible responses: each stimulus corresponds to a specific response, and the participant must first identify which stimulus appeared before selecting the matching response. The rule describing how RT grows with the number of alternatives (N) is the Hick–Hyman law, with the approximate relationship RT = a + b·log₂(N): reaction time does not grow linearly with the number of alternatives, but logarithmically.

Why it happens

The logarithmic relationship traces back to information theory: identifying "which stimulus this is" requires processing roughly log₂(N) bits of information (assuming the alternatives are equally likely), and central-processing time scales with the amount of information to be processed, not with the raw count of alternatives. That is why going from 2 to 4 alternatives adds roughly the same delay as going from 8 to 16 — rather than the latter jump costing four times as much.

Studying it

The standard approach sets up a fixed stimulus-response mapping (N lights to N keys), systematically varies N (e.g., 2, 4, 8), measures mean RT at each N, and fits a regression of RT against log₂(N); the slope reflects the processing cost per bit of information. That slope shifts with how compatible the mapping is and how practiced the participant is, so the law specifies the form of the relationship rather than a fixed numerical coefficient.

Where it stops holding

The law assumes the alternatives are equally likely, mutually independent, and that the stimulus-response mapping is already practiced and fixed. Once alternatives occur with unequal probability (one option is far more common than the rest), the mapping itself is incompatible, or the task lets the participant use context to rule out some alternatives in advance, actual RT departs from the plain log₂(N) prediction, and conditional entropy — not the raw count of alternatives — is needed to estimate the information load.

Applying it

In time-sensitive interfaces, cutting the number of alternatives a user must discriminate among at any one moment reliably lowers reaction time, but because the relationship is logarithmic, going from 8 options down to 4 buys much less than going from 4 down to 2 — so prioritize cuts that cross a power-of-two boundary (5 to 4 buys little; 4 to 2 buys a lot). How to check: record measured RT across different option-count configurations, plot RT against log₂(N), confirm whether your own interface's data follows the roughly linear relationship, and use that to judge whether further trimming the option count is still worth doing.

Related

• Same group: A8.01.1 Reaction time breaks into sensory conduction, central processing, motor conduction, and muscle activation · A8.01.2 Reaction time and movement time are two independent stages that must be measured separately · A8.01.3 Simple reaction time: one stimulus, one response · A8.01.5 Discrimination reaction time: the stimulus category must be judged before deciding whether to respond · A8.01.6 There is a floor on reaction time; anything faster is anticipation, not a genuine reaction · A8.01.7 Individual differences in reaction time exceed what most interface optimizations can gain
• Nearby: A8.02 Stimulus-response compatibility
• Search terms: choice reaction time · Hick-Hyman law · information theory reaction time