Buying voters with uncertain instrumental preferences

We analyze a vote-buying model where the members of a committee vote on a proposal important to a vote buyer. Each member incurs a privately-drawn disutility if the proposal passes. We characterize the cheapest combination of bribes that guarantees the proposal passes in all equilibria. When members vote simultaneously, the number of bribes is at least 50% larger than the number of votes required to pass the proposal (vote threshold). The number of bribes increases with the dispersion of the disutility distribution and all members are bribed with sufficient dispersion. A proportional increase in the number of members and the vote threshold leads to a less-than-proportional increase in capture cost, and the cost may increase with the vote threshold. With sequential voting and disutility distribution $U[0,1]$, all members are bribed and bribes are equal. Finally, sequential voting increases capture cost in small committees and decreases it in large committees.