A Primer on Economic Choice Automata

Abstract

This paper presents a development of the transformation semigroup of economic choice automata as a subgroup of the semigroup (monoid) of partial functions defined over the states of a finite state machine. The classes of consistency behavior considered are those rationalized by linear orders, weak orders, quasi-transitive relations and non-rationalizable path independent choice functions. For each of these classes of choice behavior, a particular class of lattice is identified as the action semigroup that drives the automaton. Given these characterizations, several features of the choice behavior are considered. In particular, the simplifying interval property of path independent choice, the importance of the distributive property of quasi-transitive rational choice in reducing the complexity of dynamic choice is addressed. Based on the algebraic structure of semiautomata implementing path independent choice functions it is possible to rank these semiautomata by the mathematical power required to implement a particular class of choice functions. This provides a means for ranking these machines by their “implementation complexity”. Dually, the computational complexity of constructing a semiautomaton that implements a particular class of choice functions is investigated. It is seen that these complexities are inversely related.