Quantum version of Aumann's approach to common knowledge: Sufficient conditions of impossibility to agree on disagree

Aumann's theorem states that if two agents with classical processing of information (and, in particular, the Bayesian update of probabilities) have the common priors, and their posteriors for a given event E are common knowledge, then their posteriors must be equal; agents with the same priors cannot agree to disagree. This theorem is of the fundamental value for theory of information and knowledge and it has numerous applications in economics and social science. Recently a quantum-like version of such theory was presented in Khrennikov and Basieva (2014b), where it was shown that, for agents with quantum information processing (and, in particular, the quantum update of probabilities), in general Aumann's theorem is not valid. In this paper we present conditions on the inter-relations of the information representations of agents, their common prior state, and an event which imply validity of Aumann's theorem. Thus we analyze conditions implying the impossibility to agree on disagree even for quantum-like agents. Here we generalize the original Aumann approach to common knowledge to the quantum case (in Khrennikov and Basieva (2014b) we used the iterative operator approach due to Brandenburger and Dekel and Monderer and Samet). Examples of applicability and non-applicability of the derived sufficient conditions for validity of Aumann's theorem for quantum(-like) agents are presented.