For incompatible observables, hidden variable interpretations need only satisfy condition (2), requiring that averages of hidden values conform to QM predictions, not (3).

The example illustrates why von Neumann’s argument is unsatisfying. Nobody disputes the move from (2) to (3) for compatible observables, i.e. those which, according to QM, are jointly measurable in one arrangement. The above choice of A, B, C, however, is such that any two of them are incompatible, i.e. are not jointly observable. For these we will not want to require any HV interpretation to meet (3), but only (2). The hidden values need not conform with (3) in general, only the averages of the