Trees and Decisions

Carlos Alós-Ferrer, University of Konstanz
Klaus Ritzberger, Institufe for Advanced Studies, Vienna

Abstract

The traditional model of sequential decision making, for instance in extensive form games, is a tree. Most texts define a tree as a connected graph without loops and a distinguished node, called the root. But a graph is not a domain for decision theory. Decision theory perceives of acts as functions from states to consequences. Sequential decisions, accordingly, get conceptualized by mappings from sets of states to sets of consequences. Thus, the question arises whether a natural definition of a tree can be given, where nodes are sets of states. We show that, indeed, trees can be defined as specific collections of sets. Without loss of generality the elements of these sets can be interpreted as representing plays. Therefore, the elements can serve as states and consequences at the same time.