I examine the effect of decision-making processes on the dynamics of bargaining over a fixed pie by comparing the share received when individuals are subject to reference-dependent preferences, loss-aversion, and probability-weighting, to the share they would receive on choosing by maximising expected utility instead. I show that: (i) reference-dependent preferences are unambiguously advantageous, (ii) loss-aversion does not have any effect, and (iii) probability-weighting is unambiguously disadvantageous. Finally, when these three features come together so that the decision-making process is described by cumulative prospect theory, then a higher share is obtained if and only if the advantage conferred by reference-dependent preferences is stronger than the disadvantage imposed by probability-weighting, and I present a precise necessary and sufficient condition that expresses this trade-off. © 2022 Elsevier B.V.