Cardinal admissibility and optimability in finite nonarchimedean decision theory

Well-known equivalences between nondominatedness or admissibility of an act and its Bayes optimality with respect to some prior probability measure are generalized to the linear nonarchimedean setting in which utilities are lexicographically ordered vectors and state probabilities are matrices that must commute multiplicatively with the matrix probabilities used in defining mixed acts to test for potential dominance. The principal result, that an act can either be made optimal or is dominated by some mixed act, but not both, is tantamount to a theorem of the alternative for doubly indexed, lexicographically ordered vectors.