We develop a modified ultimatum game, in which the proposer gives two offers, and the responder selects one offer out of the two without seeing them. Then, the selected offer becomes the relevant offer, and the unselected offer becomes the irrelevant one. Finally, the responder evaluates the fairness of the pair of offers and makes a hypothetical decision whether to accept or reject the relevant offer. For most of our subjects, the level of the irrelevant offer positively affects fairness perceptions and willingness to accept, even though the irrelevant offer cannot be accepted. The reason is that the irrelevant offer does signal the proposer’s intentions. Most responders give more weight to the relevant offer than to the irrelevant offer in evaluating fairness and in the willingness to accept. We call this effect the relevance effect. This effect is expected when considering the willingness to accept. However, it is unclear why the relevant offer should carry more weight when evaluating fairness, because the proposer makes the two offers together without knowing which one will become the relevant one. Therefore, this behavior can be considered a bias in fairness evaluations.