The game theoretic prediction for alternating offer bargaining depends crucially on how "the pie'' changes over time, and whether the proposer in an early round has ultimatum power. We experimentally study eight such games. Each game is once repeated before being followed by the next one, which defines a cycle of altogether 16 successive plays. Participants play three such cycles. There are no major experience effects but strong and reliable effects of anticipated rule changes. The latter, however, are not due to strategic considerations but rather to the social norms of fairness and efficiency."