Expressible Inspections

A decision maker needs predictions about the realization of a repeated experiment in each period. An expert provides a theory that, conditional on each finite history of outcomes, supplies a probabilistic prediction about the next outcome. However, there may be false experts without any knowledge of the data-generating process who deliver theories strategically. Hence, empirical tests for predictions are necessary. A test is manipulable if a false expert can pass the test with a high probability. Like contracts, tests have to be computable to be implemented. Considering only computable tests, we show that there is a test which passes true experts with a high probability yet is not manipulable by any computable strategy. In particular, the constructed test is both prequential and future-independent. On the other hand, any computable test is manipulable by a strategy that is computable relative to the halting problem. Our conclusion overturns earlier results that prequential or future independent tests are manipulable, and shows that computability considerations have significant effects in these problems.