When can an outsider bribe voters one by one at a negligible cost?

We discuss sequential vote-buying games, wherein an outsider, called the “principal,” sequentially attempts to influence committee members with opposite preferences regarding a project implementation before they make a decision through generalized majority voting. To characterize the equilibria, we introduce the affordably bribable groups within the members of the generalized majority, which are sets of members from whom the principal can buy votes at a cost below his gain from the project implementation. It is shown that the bribe amount and the committee decision are uniquely determined, given a sequential vote-buying game. In addition, we provide the conditions under which the principal successfully buys votes, and does so at a negligible cost, by comparing the number of affordably bribable groups with the number of members who do not belong to the generalized majority. Based on our characterization of equilibrium, we discuss the effects of committee size, derivations of simple and supermajority rules, whom to bribe, and the order of bribing.