Hick's law: decision time grows with the log of the number of choices

What it is for. Predicting how long a choice takes from the number of options, so menus, option sets, and action bars can be sized on evidence instead of taste.

The formula.

n is the number of equally likely options. a is the baseline: the simple reaction time when there is no choice to make. b is the price of growth: milliseconds per bit, fitted per person and per task, not a universal constant.

The plot. The log term against n, with n drawn to scale: bars at n = 1, 3, 7, 15, 31 stand where those numbers actually fall, so the gaps widen 2, 4, 8, 16. Each bar top carries a dot, and the dots are joined by a line that flattens as n grows: that flattening is the trend.

Bars at n = 1, 3, 7, 15, 31, spaced to scale; heights = log2(n + 1) = 1, 2, 3, 4, 5 bits. The line flattens as n grows: each doubling of n + 1 adds the same bit, so growth slows.

What kind of formula this is. An empirical law, not a theorem and not Big-O. Hick (1952) and Hyman (1953) ran choice-reaction experiments, plotted time against log2(n), and the points fell on a line: slope b, intercept a. Big-O is asymptotic order, ignoring constants and small n; Hick's law claims the actual functional form, constants included, over the tested range. It is stronger than Big-O and weaker than a theorem. The log2(n) and log2(n + 1) spellings both exist; the substance is the same.

A memorable analogy. a is the shared baseline, like a species base stat; b is the per-person growth rate, like individual values, except b is a rate, not a fixed addend: with few options a large b barely shows, with many options it dominates. n is not your attribute at all: it is the situation in front of you, the number of options on offer.

Worked case: the Lucerna reader header (counted 2026-10-25). The header carries about 14 actions. Cutting 14 to 7 moves the log term from log2(15) about 3.91 to log2(8) = 3, saving about 0.91 bits, roughly 23% of the log term. Milliseconds need a fitted b, so this stays relative, like the Fitts numbers in fitts-law.

Counterparty and boundary. Fitts is the counterparty: Fitts wants targets big, Hick wants options few, and the header is where the two negotiate. The law bends where its assumptions break: options that are not equally likely need entropy instead of log2(n); practiced users who search, chunk, or satisfice do not scan all options, so the log over-predicts.

Agent audit checklist: read this node, then check an interface against every line, and report each item with evidence.

1. Count the options sharing one decision point, and state n.
2. Compute the log term for the current n and for the proposed n, and compare alternatives by bits, not by feel.
3. Are the options equally likely? If not, the log term is the wrong input; use entropy.
4. Do users actually scan all options, or do they search, chunk, or pick the familiar one? If the latter, say so and discount the log.
5. Fitts tension: what does shrinking the option set cost in target size, screens, or hidden actions?