Many inspections are manipulable

A self-proclaimed expert uses past observations of a stochastic process to make probabilistic predictions about the process. An inspector applies a test function to the infinite sequence of predictions provided by the expert and the observed realization of the process in order to check the expert's reliability. If the test function is Borel and the inspection is such that a true expert always passes it, then it is also manipulable by an ignorant expert. The proof uses Martin's theorem about the determinacy of Blackwell games. Under the axiom of choice, there exist non-Borel test functions that are not manipulable.Forecasting, calibration, zero-sum games

The Axiomatic Structure of Empirical Content

In this paper, we provide a formal framework for studying the empirical content of a given theory. We define the falsifiable closure of a theory to be the least weakening of the theory that makes only falsifiable claims. The falsifiable closure is our notion of empirical content. We prove that the empirical content of a theory can be exactly captured by a certain kind of axiomatization, one that uses axioms which are universal negations of conjunctions of atomic formulas. The falsifiable closure operator has the structure of a topological closure, which has implications, for example, for the behavior of joint vis a vis single hypotheses. The ideas here are useful for understanding theories whose empirical content is well-understood (for example, we apply our framework to revealed preference theory, and Afriat's theorem), but they can also be applied to theories with no known axiomatization. We present an application to the theory of multiple selves, with a fixed finite set of selves and where selves are aggregated according to a neutral rule satisfying independence of irrelevant alternatives. We show that multiple selves theories are fully falsifiable, in the sense that they are equivalent to their empirical content