A recurring puzzle in bargaining experiments is that individuals under-exploit their bargaining position, compared to theoretical predictions. We conduct an experiment using two institutions: Nash demand game (NDG) and unstructured bargaining game (UBG). Unlike most previous experiments, disagreement payoffs are earned rather than assigned, and about one-fourth of the time, one bargainer's disagreement payoff is more than half the cake size (“dominant bargaining power”), so that equal splits are not individually rational. Subjects under-respond to their bargaining position most severely in the NDG without dominant bargaining power. Responsiveness increases in the UBG, but is still lower than predicted; the same is true for the NDG with dominant bargaining power. Only in the UBG with dominant bargaining power – the combination of a bargaining institution with low strategic uncertainty and elimination of the 50–50 “security blanket” – do subjects approximately fully exploit their bargaining position.