Individuals have or observe partly private information. They independently choose acts, possibly including messages. The center may also act. Individuals' utilities may depend on all acts and information, including others' private information. Are there incentives depending only on public information that make desired behavior a Bayesian equilibrium? Assume incentive payments are separable and fully transferable. Appropriate incentives exist either if the center's information--perhaps solely messages--depends stochastically, however slightly, on all relevant private information, or if individuals' relative valuations of acts, however divergent, are not too dissimilarly affected by different states of nature. More generally, we give necessary and sufficient conditions for existence whenever the strategy profile asks agents to reveal all private knowledge relevant to their beliefs about the center's information. We also develop equivalences on the possible values of private information--concepts of similarity of agent types--that are key to resolving existence questions without such responsiveness or requiring budget balance.