A dynamic and stochastic theory of choice, response time, and confidence

A Dynamic and Stochastic Theory of Choice, Response Time, and Confidence Timothy J. Pleskac (tim.pleskac@gmail.com) Indiana University, 1101 E. Tenth St. Bloomington, IN 47405 USA Jerome Busemeyer (jbusemey@indiana.edu) Indiana University, 1101 E. Tenth St. Bloomington, IN 47405 USA Abstract (where the respondent determines when to stop and decide), there is an inverse relationship between the time taken and the degree of confidence expressed in the choice (e.g.,Baranski & Petrusic, 1998; Henmon, 1911). On the other hand, during externally controlled stopping tasks (where the experimenter determines when to stop and decide) the longer people are given to make a decision the more confident they become (e.g., Irwin, Smith, & Mayfield, 1956). The purpose of this paper is to present a diffusion model that is capable of explaining all three response variables and their interrelationships using a common underlying processing mechanism. Next we give an intuitive description of how the model works using a prototypical sensory identification task. Then we formalize the model and illustrate how it simultaneously predicts the relationship between decision time and confidence for both the optional stopping and externally controlled stopping tasks. Accounting for both phenomena with a single process is an important hurdle for any model of confidence as most models of confidence can typically only account for one of these effects, but not both (Vickers, 1979). Finally, we will conclude by outlining the theoretical implications of this model. We will also offer some preliminary comparisons between the diffusion model of confidence and the Poisson race model. The three most basic performance measures used in cognitive research are choice, response time, and confidence. We present a diffusion model that accounts for all three using a common underlying process. The model uses a standard drift diffusion process to account for choice and decision time. To make a confidence judgment, we assume that evidence continues to accumulate after the choice. Judges then interrupt the process to categorize the accumulated evidence into a confidence rating. The fully specified model is shown to account qualitatively for the most important interrelationships between all three response variables found in past research. Keywords: diffusion; random walk; confidence; cognitive model; judgment. Introduction The three most important measures of cognitive performance are choice, decision time, and confidence. Signal detection theory (Green & Swets, 1966) was originally designed to explain choices and is able to also account for confidence ratings. A great limitation of this model, however, is its inability to explain decision time. Random walk/diffusion theory was introduced for this purpose (Link & Heath, 1975; Ratcliff, 1978). This theory provides an elegant explanation for both choices and response time. A great limitation of random walk/diffusion theory, up till now, is its inability to account for confidence ratings. To date only the Poisson race model has successfully accounted for all three variables (Van Zandt, 2000; Vickers, 1979; Vickers & Packer, 1982; Vickers, Smith, Burt, & Brown, 1985b). 1 The purpose of this paper is to develop a generalization of the random walk/diffusion theory to offer an alternative account of choice, decision time, and confidence. Our challenge is to explain with our diffusion model of confidence the massive amount of data that has accumulated about the complex relationships among these three measures. Here are a few. First, there is a speed/accuracy trade-off where faster choices produce higher error rates (Luce, 1986). Second, accuracy generally increases with confidence (Vickers, 1979), but judges are often overconfident (McClelland & Bolger, 1994). Finally, there is a twofold relationship between confidence and decision time. On the one hand, during optional stopping tasks A diffusion model of confidence judgments To begin, consider a standard identification task. On each trial an observer listens through headphones to either white noise or white noise plus a faint tone (signal). The observer has no way of knowing which event has occurred and must decide whether the tone is present (Yes) or if there is only white noise (No). After deciding, the observer rates her confidence in her choice by selecting one of four categories: ‘1’ for doubtful, ‘2’ for little confidence, ‘3’ for fairly confident, and ‘4’ for perfectly confident. With a faint enough signal mistakes are expected and confidence responses should be distributed over the scale. The diffusion model makes four fundamental assumptions to model observers’ Yes/No choices: (a) Evidence favoring each alternative is integrated over time during the trial; (b) The sampled evidence at each time step is subject to random fluctuations; (c) Evidence in support of one alternative Vickers and colleagues call this the accumulator model.