Self-Consistency and Common Prior in Non- Partitional Knowledge Models

In non-partitional models of knowledge with objective and subjective state spaces, the issue of self-consistency arises. The present paper defines a multigraph Gj for each player j, and also a global multigraph G. The posteriors of player j are self-consistent if and only if all cycle equations associated with cycles in Gj are satisfied. Similarly, the posteriors of all players are consistent with a common prior when all cycle equations corresponding to the cycles in G are satisfied. In particular, the self-consistency of player j is automatic when Gj is acyclic. Consistency always holds when G is acyclic, regardless of any probabilistic information. There is a simple formula to check for the acyclicity of Gj , and another formula to check for the acyclicity of G.