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 the- ory 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).