A buyer procures a good owned by a group of sellers whose heterogeneous cost of trade is private information. The buyer must either buy the whole good or nothing, and sellers share the transfer in proportion to their share of the good. We characterize the optimal mechanism: trade occurs if and only if the buyer's benefit of trade exceeds a weighted average of sellers' virtual costs. These weights are endogenous, with sellers who are ex ante less inclined to trade receiving higher weight. This mechanism always outperforms posted-price mechanisms. An extension characterizes the entire Pareto frontier.
A principal privately contracts with a set of agents who then simultaneously make a binary decision. Each contract specifies an individual allocation and the information the agent is given about a fundamental state and other agents' contracts. We study the principal's optimal scheme that induces a desired action profile as the unique rationalizable outcome. Our main result reduces this multiagent problem to a two-step procedure where information is designed agent-by-agent: the principal chooses a fundamental-state-contingent distribution over agent rankings and, separately for each agent, the agent's information about the realized ranking and fundamental states. We illustrate with a team-production application.
A principal incentivizes a team of agents to work by privately offering them bonuses contingent on team success. We study the principal's optimal incentive scheme that implements work as a unique equilibrium. This scheme leverages rank uncertainty to address strategic uncertainty. Each agent is informed only of a ranking distribution and his own bonus, the latter making work dominant provided that higher-rank agents work. If agents are symmetric, their bonuses are identical. Thus, discrimination is strictly suboptimal, in sharp contrast with the case of public contracts (Winter 2004). We characterize how agents' ranking and compensation vary with asymmetric effort costs.