CONSISTENCY IMPLIES THAT PLAUSIBILITY HAS A MASS FUNCTION
Peter A. Streufert · 2012
We formulate and prove a new fundamental theorem about Kreps-Wilson consistency. First, we derive from an arbitrary assessment its implied plausibility (i.e. innite relative likelihood) relation among the game's nodes. Typically such a plausibility re- lation is incomplete (i.e. fails to compare all nodes). Second, since nodes can be specied as sets of actions via Streufert (2012a), we introduce the concept of representing a completion of a plausibility relation by the nodal sums of a mass (i.e. density) function assign- ing plausibility numbers to the game's actions. Finally, we discover that the consistency of an assessment implies that its plausibility relation has a completion represented by a mass function. We prove this by re-using math from the early foundations of proba- bility theory. This theorem leads to a number of corollaries. First, we are able to formalize in two new ways that consistency species that zero-probability agents reason that past zero-probability actions were played independently. Second, we identify and repair a non- trivial gap in a Kreps-Wilson proof and then clarify two algebraic (i.e. non-topological) characterizations of consistency from the lit- erature. Third, we discover a particularly simple characterization of consistency for degenerate-support (i.e. pure-strategy and sure- belief) assessments. All the paper's proofs are accessible in that they require nothing more than linear algebra.