In a variety of settings, budgets are set by a committee that interacts repeatedly over many budget cycles. To capture this, we study a model of repeated multilateral bargaining by a budget committee. Our focus is on the transition of agenda setting power from one cycle to the next, and how such considerations affect bargaining and coalition formation over time. Specifically, we compare a rule that approximates the budget process in many parliamentary democracies in which a vote of confidence is traditionally attached to each budget proposal, and a rule that approximates the budget process in congressional systems where party leadership must maintain the support of a majority of other legislators to hold onto power. As is standard in the literature, we use stationary equilibrium refinements to make predictions about behavior in our environments. In a controlled laboratory experiment, we find no support for the standard equilibrium refinements used in the literature. In sharp contrast to the theoretical predictions, in the experiment, both rules give rise to stable and persistent coalitions in terms of coalition size, identity, and shares of coalition partners and feature high persistence of agenda-setter power. Our results call into question the validity of restricting attention to history independent strategies in dynamic bargaining games. We conclude by showing that weakening the standard equilibria concepts to allow players to condition on one piece of history (the most recent deviator) is enough to generate equilibria which are consistent with outcomes and behavior observed in the experiments.