Ultimatums in two-person bargaining with one-sided uncertainty: Offer games

In the ultimatum game with complete information a Sender proposes a division of a given amount of surplus (“pie”), which a Receiver can either accept (payoffs are distributed according to the Sender's proposal) or reject (both players earn nothing). We study another version of the ultimatum game under incomplete informaton in which the pie is drawn randomly from a commonly known distribution, the Sender knows the exact size of the pie, but the Receiver only knows her share of the pie, not the residual share requested by the Sender. The basic results are that (1) as the support of the pie distribution increases in a mean-preserving spread, the Senders make lower offers that the Receivers are less likely to reject, (2) for a given support, Senders tend to offer a lower proportion of the pie to the Receivers as the pie size grows larger, and (3) although knowing only their share of the pie, Receivers estimate its size quite accurately.