In this paper, we study the (symmetric) equilibria of a model of multilateral bargaining where players are heterogeneous regarding their time preferences, and make costly efforts at the beginning of the process in order to in.uence their probabilities of being the proposer for all stages of the negotiation process. We analyse whether the optimality of the unanimity rule (as the voting rule minimizing the social cost resulting from the agents'willingness to buy influence) characterised in Yildirim (2007) extends to the present situation. In the case of weakly heterogeneous agents, we show that k-majority rules may actually become strictly optimal. Then we provide numerical examples that suggest that there are situations where each type of voting rule (unanimity and strict k-majority) may be socially optimal.