Decision Theory in an Algebraic Setting
This work provides a theoretical extension of decision theory for researchers in mathematical foundations, but it is incremental as it builds on existing algebraic frameworks.
The paper tackles the problem of generalizing decision theory by replacing the algebra of events with a finite distributive lattice and probability measures with lattice valuations, resulting in a lattice structure for acts and analysis of comparisons without common conditions.
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.