Group-living animals routinely have to reach a consensus decision and choose between mutually exclusive actions in order to coordinate their activities and benefit from sociality [1] and [2]. Theoretical models predict “democratic” rather than “despotic” decisions to be widespread in social vertebrates, because they result in lower “consensus costs”—the costs of an individual foregoing its optimal action to comply with the decision—for the group as a whole [1] and [3]. Yet, quantification of consensus costs is entirely lacking, and empirical observations provide strong support for the occurrence of both democratic and despotic decisions in nature [1], [4] and [5]. We conducted a foraging experiment on a wild social primate (chacma baboons, Papio ursinus) in order to gain new insights into despotic group decision making. The results show that group foraging decisions were consistently led by the individual who acquired the greatest benefits from those decisions, namely the dominant male. Subordinate group members followed the leader despite considerable consensus costs. Follower behavior was mediated by social ties to the leader, and where these ties were weaker, group fission was more likely to occur. Our findings highlight the importance of leader incentives and social relationships in group decision-making processes and the emergence of despotism.