Journal of Mathematical Psychology 54 (2010) 338–340 Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp Theoretical note The exponential learning equation as a function of successful trials results in sigmoid performance Nathaniel Leibowitz, Barak Baum, Giora Enden, Amir Karniel ∗ Department of Biomedical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel article info Article history: Received 5 November 2009 Received in revised form 9 January 2010 Available online 6 March 2010 Keywords: Learning Success rate Exponential learning curve Sigmoid learning curve abstract While the exponential learning equation, indicating a gradually diminishing improvement, is one of the standard equations to describe learning, a sigmoid behavior with initially increasing then decreasing improvement has also been suggested. Here we show that the sigmoid behavior is mathematically derived from the standard exponential equation when the independent variable of the equation is restricted to the successful trials alone. It is suggested that for tasks promoting success-based learning, performance is better described by the derived sigmoid curve. © 2010 Elsevier Inc. All rights reserved. 1. Introduction The exponential learning equation has been derived analytically by several researchers (Estes, 1950; Hull, 1943; Thurstone, 1919) and is one of the standard equations to describe the improvement in the performance of tasks with practice (Heathcote, Brown, & Mewhort, 2000; Ritter & Schooler, 2001): P n = P ∞ − (P ∞ − P 0 ) · e −α·n , where n denotes trial number, P n the performance measure at trial n, and p 0 , p ∞ the initial and asymptotic performance, respectively and α is a constant rate coefficient. The concavity of P n implies a monotonically decreasing improvement (Δ n = P n −P n−1 < Δ n−1 ). However, a sigmoid behavior in which the improvement initially increases then decreases has been persistently suggested based on either empirical observations (Culler, 1928; Culler & Girden, 1951; Gallistel, Fairhurst, & Balsam, 2004; Woodworth, 1938) or on analytical derivation from assumptions on the underlying learning process (Gulliksen, 1934, 1953; Mazur & Hastie, 1978; Newell, Liu, & Mayer-Kress, 2001; Thurstone, 1930). Here we note that by an alternative interpretation of the independent parameter n, this presumably contradictory observation of sigmoid behavior can in fact be predicted by the traditional exponential equation. In conventional usage of the exponential equation, where the independent parameter n equals the number of trials, all trials ∗ Corresponding address: Department of Biomedical Engineering, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel. E-mail address: akarniel@bgu.ac.il (A. Karniel). are assumed to equally affect the learning process. This approach is justified for certain applications of the learning equation. For instance, when describing improvement in the response time of well-practiced tasks, such as in the frequently cited cigar- rolling study (Crossman, 1959), it is reasonable to attribute equal weights to all trials. However, applying the equation to describe the improvement in the success rate of a task requires a clear distinction between successful and failed trials. When the successful responses are a small fraction of all possible responses, a successful response may provide significantly more information to the learner than a failed response. In the extreme case, when the range of possible responses is very large, a single failed response may provide little (if any) information for improving performance. We therefore propose that for such tasks, the exponential learning equation should be re-defined in terms of the number of successful trials only, rather than the total number of trials, and we show in the next section that this modification leads to the classical sigmoid behavior. In a subsequent section the general case in which the weighted average of success and of failure in facilitating learning is addressed, demonstrating a gradual shift towards a sigmoid function as the weight of successful trials is increased. 2. Success-based learning Consider a behavioral non-trivial task (i.e. one which requires numerous repeated trials to master) in which each trial is either a success or a failure. In such a case one needs to average the success over a group of trials in order to obtain a non-binary measure of success. If the trial in the standard learning curve is replaced by a block of trials, we obtain P n = P ∞ − (P ∞ − P 0 ) · e −α·b·n , 0022-2496/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2010.01.006