A Mechanism for Value-Sensitive Decision-Making
(Left) Bifurcation set as a function of and , for fixed . This generalises the result of Figure 2, for which . The grey region corresponds to parameters where the decision dynamics have a single stable attractor (pre-bifurcation), whereas the white region corresponds to those having two stable attractors and one saddle node (post-bifurcation). Sample phase-portraits illustrate how the positions of these fixed points change according to and . Plots (a) and (b) illustrate the results of Figure 2, in which . Increasing moves the stable attractor towards the superior alternative in the pre-bifurcation case (see plot (c)), although it may still correspond to a population state in which threshold is reached for neither alternative; whereas increasing in the post-bifurcation case moves the saddle point towards the inferior alternative, thereby increasing the basin of attraction for the superior alternative (see plot (d)). Thus for a decision with given that is too low to precipitate a threshold decision, increasing precipitates a decision, in which the more valuable alternative is more likely to be selected. (Middle) The relationship between and the minimum required for a unique attractor for the best alternative depends on . (Right) The relationship between and the minimum required for a single alternative to unambiguously be considered the best converges on a linear relationship, with slope determined by . This is similar to Weber's law of just noticeable difference, observed in psychological studies, with determining the Weber coefficient.