The accumulation of findings that most responders in the ultimatum game reject unfair offers provides evidence that humans are driven by social preferences such as preferences for fairness and prosociality. On the other hand, if and how the proposer's behavior is affected by social preferences remains unelucidated. We addressed this question for the first time by manipulating the knowledge that the proposer had about the responder's belief concerning the intentionality of the proposer. In a new game called the "ultimatum game with ambiguous intentions of the proposer (UGAMB)," we made the intentionality of the proposer ambiguous to the recipient. We expected and found that the proposer would make more unfair offers in this new game than in the standard ultimatum game. This expectation can be derived from either the preference-based model or the strategy model of the proposer's giving decision. The additional finding that more unfair giving in the UGAMB was not mediated by the proposer's expectation that the recipient would be more willing to accept unfair offers provided support for the preference-based model. Using a psychological measure of cognitive control, the preference-based model received additional support through a conceptual replication of the previous finding that cognitive control of intuitive drive for prosociality in the dictator game, rather than mind reading in the ultimatum game, is responsible for the difference in giving between the two games.