Is there a random walk theory that can account for situations with more than two choices?

Question

In the article "Two-stage Dynamic Signal Detection: A Theory of Choice, Decision Time, and Confidence" from 2010 by Pleskac and Busemeyer, a random walk model is presented for situations where a discrete choice is made (that is, signal present/not present, yes/no, 1/0 et cetera). Here, if there is no time pressure involved, a threshold value is set which has to be reached before an answer is given. Contrary, if there is a time pressure involved, the value at the last time point is used to decide which answer is given, even if the threshold point hasn't been reached at that time.

I wonder whether there are any extension of this model which can incorporate situations with more than two choices (from 3 alternatives and up to a fully continius choice space). I guess you just could incorporate more dimensions (the model referred to above is two-dimensional), but if the answers are non-independent (for example, when using a rating scale), I guess the random walk function would have to be fairly complicated. That is, if strong evidence is accumulated for number 3, how much evidence should be added to 4 and 2? Should evidence be subtracted from 1 and 5? 

Answer 1

Instead of having a single integrator with two bounds for two choices (symmetric random walk model), you can have many competing integrators each with a bound (race model). For example, see Fig 2. of Gold and Shadlen 2007 and references therein.

As for the continuous choices case, it is important to understand a limit of discrete choices can be very different from a continuous choice. For such a limit to make sense, there should be a notion of similarity between the choices, and I don't think those race models would generalize easily. A high dimensional continuous attractor dynamics is more likely to support such theory.

Answer 2

Diederich & Busemeyer (2003) presented a diffusion model for three choice alternatives (p. 314). The paper is a tutorial for calculating diffusion models with (discrete) matrix methods. The extension to three choice alternatives is reached by defining a two-dimensional diffusion process on a triangular plane (state space).

Recently, Wollschläger & Diederich (2012) presented the 2N-ary choice tree model which models multi-alternative choices by random walks on decision trees.

References
Diederich, A. & Busemeyer J. R. (2003). Simple matrix methods for analyzing diffusion models of choice probability, choice response time, and simple response time. Journal of Mathematical Psychology, 47(3), 304-322. doi:10.1016/S0022-2496(03)00003-8 (preprint pdf)

Wollschläger LM and Diederich A (2012) The 2N-ary choice tree model for N-alternative preferential choice. Front. Psychology 3:189. doi: 10.3389/fpsyg.2012.00189

Answer 3

Bogacz et al. (2006) provide the most comprehensive overview of models in this domain. This includes comparisons of the Drift Diffusion Model (Ratcliff, 1978), Ornstein–Uhlenbeck (O-U) Model (e.g. Busemeyer & Townsend, 1993; "Decision Field Theory"), race models without inhibition (e.g. Vickers, 1970), and race models with inhibition (e.g. Usher & McClelland, 2001; "leaky competing accumulator model") and others.

While diffusion models are limited to 2 alternatives (e.g. DDM, DFT), race models are not. (Note that diffusion models are essentially the same as random walk models, except that random walks involve discrete steps, while diffusion models are continuous). The race models with inhibition reviewed by the authors are computationally reducible to DDM, while race models without inhibition are different, and generally not preferable.

Beck et al. (2008) also provide a model that accommodates continuous choice, rather than a limited number of alternatives. It's relationship to other models is unclear.

So yes, there are many similar models that account for more than 2 alternatives. They are not technically random walk models, because a random walk only allows for 2 bounds. But they are computationally equivalent.

Beck, J. M., Ma, W. J., Kiani, R., Hanks, T., Churchland, A. K., Roitman, J., ... & Pouget, A. (2008). Probabilistic population codes for Bayesian decision making. Neuron, 60(6), 1142-1152. PDF

Bogacz, R., Brown, E., Moehlis, J., Holmes, P., & Cohen, J. D. (2006). The physics of optimal decision making: a formal analysis of models of performance in two-alternative forced-choice tasks. Psychological review, 113(4), 700. PDF

Busemeyer, J. R., & Townsend, J. T. (1993). Decision field theory: a dynamic-cognitive approach to decision making in an uncertain environment. Psychological review, 100(3), 432. PDF

Ratcliff, R. (1978). A theory of memory retrieval. Psychological review, 85(2), 59. PDF

Usher, M., & McClelland, J. L. (2001). The time course of perceptual choice: the leaky, competing accumulator model. Psychological review, 108(3), 550. PDF

Vickers, D. (1970). Evidence for an accumulator model of psychophysical discrimination. Ergonomics, 13(1), 37-58.