We present a dynamical systems analysis of a decision-making mechanism inspired by collective choice in house-hunting honeybee swarms, revealing the crucial role of cross-inhibitory 'stop-signalling' in improving the decision-making capabilities. We show that strength of cross-inhibition is a decision-parameter influencing how decisions depend both on the difference in value and on the mean value of the alternatives; this is in contrast to many previous mechanistic models of decision-making, which are typically sensitive to decision accuracy rather than the value of the option chosen. The strength of cross-inhibition determines when deadlock over similarly valued alternatives is maintained or broken, as a function of the mean value; thus, changes in cross-inhibition strength allow adaptive time-dependent decision-making strategies. Cross-inhibition also tunes the minimum difference between alternatives required for reliable discrimination, in a manner similar to Weber's law of just-noticeable difference. Finally, cross-inhibition tunes the speed-accuracy trade-off realised when differences in the values of the alternatives are sufficiently large to matter. We propose that the model, and the significant role of the values of the alternatives, may describe other decision-making systems, including intracellular regulatory circuits, and simple neural circuits, and may provide guidance in the design of decision-making algorithms for artificial systems, particularly those functioning without centralised control.