Hick's law

Hick's law, or the Hick-Hyman law, is a human-computer interaction model that describes the time it takes for a user to make a decision as a function of the possible choices he or she has. Given n equally probable choices, the average reaction time T required to choose among them is approximately


 * $$T = b \log_{2}(n + 1)$$

where b is a constant that can be determined empirically by fitting a line to measured data. According to Card, Moran, and Newell (1983), the +1 is "because there is uncertainty about whether to respond or not, as well as about which response to make." The law can be generalized in the case of choices with unequal probabilities pi of occurring, to


 * $$T = b H$$

where H is the information-theoretic entropy of the decision, defined as


 * $$H = \sum_i^n p_i \log_{2}(1/p_i + 1)$$

Hick's law is similar in form to Fitts' law. Intuitively, one can reason that Hick's law has a logarithmic form because people subdivide the total collection of choices into categories, eliminating about half of the remaining choices at each step, rather than considering each and every choice one-by-one, requiring linear time.

Hick's law has been shown to apply in experiments where the user is presented with n buttons, each having a light bulb beside them. One light bulb is randomly lit up, after which the user must press the corresponding button as quickly as possible. Obviously, the decision to be made here is very simple, requiring little conscious thought.

Hick's law is sometimes cited to justify menu design decisions (for an example, see ). However, applying the model to menus must be done with care. For example, to find a given word (e.g. the name of a command) in a randomly ordered word list (e.g. a menu), scanning of each word in the list is required, consuming linear time, so Hick's law does not apply. However, if the list is alphabetical, the user will likely be able to use a subdividing strategy that may well require logarithmic time. Of course, well-designed submenus can allow for automatic subdivision.

Yet another situation is when the user does not know the exact name of the command they seek in a menu, but would likely recognize it if they saw it. In this case, the user may or may not be able to use a subdividing search strategy, depending in part on how menu items are categorized and how well the user can use categories to speed their search.

For Hick's law and Fitts' law considerations in the context of menu and submenu design, see Landauer and Nachbar (1985).