## 1. Introduction

One of the key phenomena in response time (RT) research is the speed-accuracy trade-off, by which a decision maker can speed up at the expense of accuracy and become more accurate at the expense of speed (Bogacz, Wagenmakers, Forstmann, & Nieuwenhuis, 2010; Schouten & Bekker, 1967; Wickelgren, 1977). The interdependence of RT and accuracy implies that people can be accurate and slow in one situation, yet fast and inaccurate in another, although their efficiency in information processing does not change. The speed-accuracy trade-off therefore frustrates a straightforward interpretation of RT in terms of cognitive processing time and forces researchers to consider RT and accuracy jointly.

Most models that account for the speed-accuracy trade-off, including most sequential sampling models, implicitly assume that the speed-accuracy trade-off is a continuous function. This assumption implies that a participant who is responding accurately on a certain task can gradually increase speed at the cost of gradual decreases in accuracy, until speed reaches ceiling and accuracy is at chance level (i.e., fast guessing). Here, we challenge this assumption and hypothesize that with increasing pressure to respond quickly, relatively accurate behavior suddenly collapses into guessing behavior, without going through all the intermediate stages between accurate responding and guessing.

To account for this discontinuous shift in performance, we introduce a phase transition model for the speed-accuracy trade-off. The model postulates that guessing and stimulus-controlled responding are irreconcilable modes of processing. This means that when experimental settings continuously change and force people to switch from one mode of processing to the other, this switch will be abrupt. When participants are, for example, forced to speed up over trials (and become less careful), at first they will be able to persist in fairly accurate responding. However, with a gradual increase in speed stress, performance will at some point break down completely and participants abruptly resort to fast guessing. Our model predicts a similar abrupt switch when the experimental conditions gradually encourage participants to stop guessing and be more careful (and respond more slowly).

Our phase transition model finds its roots in Ollman's fast guess model (Ollman, 1966). However, our model offers a more dynamic account of the speed-accuracy trade-off and allows for a connection to sequential sampling models of RT such as Ratcliff's diffusion model (Ratcliff, 1978). The phase transition model has the form of a cusp model from catastrophe theory. Catastrophe theory is a mathematical theory that applies to dynamic systems in which continuous changes of environmental variables lead to sudden changes in observed behavior (e.g., Zeeman, 1976). From this model, we derive two signature predictions of the phase transition model: *hysteresis* and *bimodality*. We test these two predictions in two experiments.

The outline of this article is as follows: In the first section, we discuss sequential sampling models and varied state models of the speed-accuracy trade-off. In the second section, we introduce the phase transition model. In the third section, we explain the two experiments that test the predictions of our model. Next, the experimental data are described and discussed by means of quantitative models (i.e., a hidden Markov model and our cusp model).