Daily life confronts us with many choice situations in which three or more options are available. What should we do? Take the best? Eliminate all but one? There are numerous empirical studies and theoretical approaches on multi-alternative choice situations with very different settings and foci (e.g., Hensher et al., 2015, Louviere et al., 2004 for an overview of designs and models). The current paper is on best–worst scenarios only. In a best–worst task, a person when faced with three or more choice alternatives is required to identify the best and the worst choice option. This method received increasing theoretical and practical interest over the last decade because it has a number of advantages over the traditional choice tasks in which the decision maker is typically asked to choose only the best alternative (Marley & Louviere, 2005 p. 464). It has been applied to choosing among consumer products, health services and treatments, perceptual stimuli, and more (e.g., Diederich et al., 2012, Hawkins et al., 2014a, Hawkins et al., 2014b, Soekhai et al., 2021). Note that, “best” and “worst” refer to any terms that define extremes on a subjective continuum, for instance, largest and smallest (Louviere et al., 2015 p. 6).

The majority of theoretical best–worst approaches is based on random ranking and random utility, joint and sequential, and ratio scales (Marley & Louviere, 2005 p. 465) and focusses on axiomatizations of best–worst tasks (Louviere et al., 2015, Marley, 2020). There are hardly any models that take the dynamics of decision-making into account. An exception is the work by Hawkins et al., 2014a, Hawkins et al., 2014b who extended the linear ballistic accumulator (LBA) model to best–worst choice data, called parallel LBA model. It assumes a race between independent accumulations associated with each choice option in the choice set. A so-called drift rate d of an accumulator represents the utility for a given choice option. With n choice options in the choice set, n accumulators with drift rate di for option i,i=1,2,…,n are in the race to decide the best option. In a parallel race, n accumulators with drift rates 1/di for the according option i are in the race to decide the worst option.

The goal of this work is to introduce another dynamic model for the best–worst choice task as well as for ranking tasks that is based on a multi-dimensional diffusion process. The approach is a natural modification of the cube model that was previously studied in Mallahi-Karai and Diederich, 2019, Mallahi-Karai and Diederich, 2021 in the context of multi-alternative choice situations with choosing the best option only. Note the that primary goal here is to convey the basic concept and ideas.

The paper is organized as follows. Section 2 sets the notation and introduces the mathematical objects used in the remainder of the paper. In Section 3 we briefly describe the cube model and recall definitions of complete and partial rankings. In Section 4 we will give a detailed description of the cube model for ranking. Finally, in Section 6 we will discuss some problems related to the proposed model that will be studied in future work.