Sequential decisions with tests

We consider a principal-agent problem where the principal wishes to be endorsed by a sequence of agents, but cannot truthfully reveal type. In the standard “herding” model, the agents learn from each other's decisions, which can lead to cascades on a given decision when later agents' private information is swamped. We augment the standard model to allow the principal to subject herself to a test designed to provide public information about her type. She must decide how tough a test to attempt from a continuum of test types, which involves trading off the higher probability of passing an easier test against the greater impact from passing a tougher test. We find that the principal will always choose to be tested, and will prefer a tough test to a neutral or easy one.