Hick’s law

The Paradox of Choice – Why More Is Less is a 2004 book by American psychologist Barry Schwartz. In the book, Schwartz argues that eliminating consumer choices can greatly reduce anxiety for shoppers.

Autonomy and Freedom of choice are critical to our well being, and choice is critical to freedom and autonomy. Nonetheless, though modern Americans have more choice than any group of people ever has before, and thus, presumably, more freedom and autonomy, we don’t seem to be benefiting from it psychologically.

— quoted from Ch.5, The Paradox of Choice, 2004

Hick’s law, or the Hick–Hyman Law, named after British and American psychologists William Edmund Hick and Ray Hyman, describes the time it takes for a person to make a decision as a result of the possible choices he or she has: increasing the number of choices will increase the decision time logarithmically. The Hick–Hyman law assesses cognitive information capacity in choice reaction experiments. The amount of time taken to process a certain amount of bits in the Hick–Hyman law is known as the rate of gain of information.

Hick’s law is sometimes cited to justify menu design decisions. 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 and the user knows the name of the command, he or she may be able to use a subdividing strategy that works in logarithmic time.[1]

Studies suggest that the search for a word within a randomly ordered list – in which the reaction time increases linearly according to the number of items – does not allow for the generalization of the scientific law, considering that, in other conditions, the reaction time may not be linearly associated to the logarithm of the number of elements or even show other variations of the basic plane.

Exceptions to Hick’s law have been identified in studies of verbal response to familiar stimuli, where there is no relationship or only a subtle increase in the reaction time associated with an increased number of elements,[4] and saccade responses, where it was shown that there is either no relationship,[5] or a decrease in the saccadic time with the increase of the number of elements, thus an antagonistic effect to that postulated by Hick’s law.[6]

The generalization of Hick’s law was also tested in studies on the predictability of transitions associated with the reaction time of elements that appeared in a structured sequence.[7] [8] This process was first described as being in accordance to Hick’s law,[9] but more recently it was shown that the relationship between predictability and reaction time is sigmoid, not linear associated with different modes of action. [10]

The power law of practice states that the logarithm of the reaction time for a particular task decreases linearly with the logarithm of the number of practice trials taken. It is an example of the learning curve effect on performance. It was first proposed as a psychological law by Newell & Rosenblom.[1] Delaney et al. showed that the power law fit better than an exponential if the analysis was performed across strategies, for a mental arithmetic task.[2]

However, subsequent research by Heathcote, Brown, and Mewhort suggests that the power function observed in learning curves that are averaged across participants is an artifact of aggregation.[3] Heathcote et al. suggest that individual-level data is better fit by an exponential function and the authors demonstrate that the multiple exponential curves will average to produce a curve that is misleadingly well fit by a power function.

The power function is based on the idea that something is slowing down the learning process; at least, this is what the function suggests. Our learning does not occur at a constant rate according this function; our learning is hindered. The exponential function shows that learning increases at a constant rate in relationship to what is left to be learned. If you know absolutely nothing about a topic, you can learn 50% of the information quickly, but when you have 50% less to learn, it takes more time to learn that final 50%.

Research by Logan suggests that the instance theory of automaticity can be used to explain why the power law is deemed an accurate portrayal of reaction time learning curves.[4] A skill is automatic when there is one step from stimulus to retrieval. For many problem solving tasks, reaction time is related to how long it takes to discover an answer, but as time goes on, certain answers are stored within the individual’s memory and they have to simply recall the information, thus reducing reaction time. This is the first theory that addresses the why of the power law of practice.

Power function:

RT = aPb + c

Exponential function:

RT = aeb(P-1) + c

Where

RT = Trial Completion Time
P = Trial Number, starting from 1 (for exponential functions the P-1 argument is used)
a, b, and c, are constants

Practice effects are also influenced by latency. Anderson, Fincham, and Douglass looked at the relationship between practice and latency and people’s ability to retain what they learned. As the time between trials increases, there is greater decay. The latency function relates to the forgetting curve.[5]

Latency Function:

latency = A + B*Td

Where

A = asymptotic latency B = latency that varies T = time between introduction and testing d = decay rate

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