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Individual leaders are often responsible for high-stakes bargaining between larger aggregates, such as nations and firms. While these leaders may have input from others, the ultimate decision in these bargaining games usually falls upon the leaders themselves. So, it remains an important question as to whether leaders can effectively aggregate the wisdom of their advisers. Here, we address this question by conducting the first large-scale experiment on deliberation and collective decision-making in ultimatum bargaining games. In these experiments, we compare individuals, to groups where each member has an equal say, to groups with a leader who makes a choice on behalf of the entire group. We find a ``wisdom of crowds" effect, whereby deliberating groups outperform individual bargainers across a number of dimensions, even when the group's ultimate decision is determined by a single group leader. This occurs because leaders' decisions tend to reflect the views of their entire group.