The ultimatum game, by its all-or-nothing nature, makes it difficult to discern what kind of preferences may be generating choices. We explore a game that convexifies the decisions, allowing us a better look at the indifference curves of bargainers while maintaining the subgame-perfect equilibrium. We conclude that bargainers' preferences are convex and regular but not always monotonic. Money-maximization is the sole concern for about half of the subjects, while the other half reveal a preference for fairness. We also found, unexpectedly, the importance of risk aversion among money-maximizing proposers, which in turn generates significant bargaining power for fair-minded responders.