Implementation with evidence

We generalize the canonical problem of Nash implementation by allowing agents to voluntarily provide discriminatory signals, i.e. evidence. Evidence can either take the form of hard information or, more generally, have differential but non-prohibitive costs in different states. In such environments, social choice functions that are not Maskin-monotonic can be implemented. We formulate a more general property, evidence-monotonicity, and show that this is a necessary condition for implementation. Evidence-monotonicity is also sufficient for implementation in economic environments. In some settings, such as when agents have small preferences for honesty, any social choice function is evidence-monotonic. Additional characterizations are obtained for hard evidence. We discuss the relationship between the implementation problem where evidence provision is voluntary and a hypothetical problem where evidence can be chosen by the planner as part of an extended outcome space.