Additive Plausibility Characterizes the Supports of Consistent Assessments

We introduce three definitions. First, we let a "basement" be a set of nodes and actions that supports at least one assessment. Second, we derive from an arbitrary basement its implied "plausibility" (i.e. infinite-relative-likelihood) relation among the game's nodes. Third, we say that this plausibility relation is "additive" if it has a completion represented by the nodal sums of a mass function defined over the game's actions. This last construction is built upon Streufert (2012)'s result that nodes can be specified as sets of actions. Our central result is that a basement has additive plausibility if and only if it supports at least one consistent assessment. The result's proof parallels the early foundations of probability theory and requires only Farkas' Lemma. The result leads to related characterizations, to an easily tested necessary condition for consistency, and to the repair of a nontrivial gap in a proof of Kreps and Wilson (1982).