The ultimatum game has been the primary tool for studying bargaining behavior in recent years. However, not enough information is gathered in the ultimatum game to get a clear picture of responders’ utility functions. We analyze a convex ultimatum game in which responders’ can “shrink” an offer as well as to accept or reject it. This allows us to observe enough about responders’ preferences to estimate utility functions. We then successfully use data collected from convex ultimatum games to predict behavior in standard games. Our analysis reveals that rejections can be “rationalized” with neo-classical preferences over own- and other-payoff that are convex, nonmonotonic, and regular. These findings present a precise benchmark for models of fairness and bargaining.