•A multilateral bargaining mechanism is studied that relaxes the institutional feature of ‘every responder has veto power’.•The mechanism has good efficiency properties in the setting of strictly supermodular games.•Provided bargaining frictions are not too high, efficient equilibria in pure stationary strategies exist.•All such equilibria are payoff equivalent in the limit as bargaining frictions vanish.•The limiting payoff is the core-constrained Nash Bargaining Solution. A new feature pertaining to proposer's ability to implement offers is introduced in the extensive form bargaining mechanism studied in Okada (1996). This mechanism is used to analyze the coalitional setting of strictly supermodular games. The new feature in the mechanism is that the proposer has a choice to implement his proposal with any subset of responders who have accepted it. Thus the institutional feature of ‘every responder has veto power’ is relaxed here. It is shown that for all sufficiently high discount factors δ, there exists an efficient subgame perfect equilibrium in pure stationary strategies (SSPE) whose limiting outcome is the core-constrained Nash Bargaining Solution. Moreover, all efficient SSPE are payoff-equivalent in the limit as δ→1.