Although conflicts in bargaining have attracted a lot of attention in the literature, situations in which bargainers have to share the product of their performance have been rarely investigated theoretically and empirically. Here, by decomposing the well-known overplacement effect, we show that two types of biases can lead to conflict in these situations: players might be overconfident in their own production (overconfidence bias) and / or underestimate the production of others (other-underestimation bias). To quantify these biases, we develop a novel experimental setting using a psychophysically controlled production task within a bargaining game. In comparison to Bayesian agents, participants tend to disagree too often, partly because they exhibit both cognitive biases. We test interventions to mitigate these biases, and are able to increase settlements mainly by reducing the other-underestimation bias. Our approach illustrates how combining psychophysical methods and economic analyses could prove helpful to identify the impact of cognitive biases on individuals' behavior.