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.

An agent’s preferences depend on an ordered parameter or type. We characterize the set of utility functions with single-crossing differences (SCD) in convex environments. These include preferences over lotteries, both in expected utility and rank-dependent utility frameworks, and preferences over bundles of goods and over consumption streams. Our notion of SCD does not presume an order on the choice space. This unordered SCD is necessary and sufficient for “interval choice” comparative statics. We present applications to cheap talk, observational learning, and collective choice, showing how convex environments arise in these problems and how SCD/interval choice are useful. Methodologically, our main characterization stems from a result on linear aggregations of single-crossing functions.

When does society eventually learn the truth, or take the correct action, via observational learning? In a general model of sequential learning over social networks, we identify a simple condition for learning dubbed excludability . Excludability is a joint property of agents' preferences and their information. We develop two classes of preferences and information that jointly satisfy excludability: (i) for a one‐dimensional state, preferences with single‐crossing differences and a new informational condition, directionally unbounded beliefs; and (ii) for a multi‐dimensional state, intermediate preferences and subexponential location‐shift information. These applications exemplify that with multiple states, “unbounded beliefs” is not only unnecessary for learning, but incompatible with familiar informational structures like normal information. Unbounded beliefs demands that a single agent can identify the correct action. Excludability, on the other hand, only requires that a single agent must be able to displace any wrong action, even if she cannot take the correct action.