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 who have no knowledge of the data-generating process and 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 that 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. Alternatively, 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.