Decision Theory in an Algebraic Setting

Abstract

In decision theory an act is a function from a set of conditions to the set of real numbers. The set of conditions is a partition in some algebra of events. The expected value of an act can be calculated when a probability measure is given. We adopt an algebraic point of view by substituting the algebra of events with a finite distributive lattice and the probability measure with a lattice valuation. We introduce a partial order on acts that generalizes the dominance relation and show that the set of acts is a lattice with respect to this order. Finally we analyze some different kinds of comparison between acts, without supposing a common set of conditions for the acts to be compared.