Decision Theory in an Algebraic Setting

TL;DR

This paper develops an algebraic framework for decision theory by replacing classical event algebras with finite distributive lattices and analyzing acts within this structure, generalizing traditional comparison methods.

Contribution

It introduces a lattice-based algebraic approach to decision theory, extending the classical model by using lattice valuations and a generalized order on acts.

Findings

Acts form a lattice under the new partial order.

The generalized dominance relation extends classical comparisons.

Different act comparison methods are analyzed without common conditions.

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.