Two Motives for Verification in Information Cascades

We study sequential social learning when agents can pay to conduct a publicly observed investigation before acting. The baseline test is one-sided: success conclusively establishes one state, whereas failure is ordinarily inconclusive; success also gives the investigator a discovery reward. Investigation therefore has two private returns. Its diagnostic value is highest near the action threshold and vanishes at sufficiently optimistic beliefs, while the expected discovery reward is weakly increasing. The equilibrium investigation set has at most two components and can be disconnected, with a central diagnostic region and a separate high-belief, reward-driven region. Because investigation is selected on private information, the attempt itself is informative. The exact public transition map shows that a failed investigation can raise public belief when favorable selection outweighs the adverse outcome. Nevertheless, a positive chance of proof at a given history does not guarantee eventual discovery: with strictly positive cost, the total number of attempts is finite almost surely under either state, and a failure can move beliefs into an absorbing cascade trap. The two-component geometry persists for sufficiently small false-positive rates and can also arise with heterogeneous private costs. The stopping result, however, relies on conclusive evidence and a positive lower bound on costs.