Lattices, Lotteries and Choice

Abstract

This paper addresses the classic Arrow-Pratt model of choice under uncertainty, relaxing one important condition, and using different tools. The condition relaxed is the single variable in preferences, by allowing for imperfect substitutability in preferences. The tools used are not topological but lattice theoretic. It is argued that when goods are imperfectly substitutable in preferences, not only attitudes to risk but also underlying preferences determine choice and therefore, a complex interaction between preferences and attititude to risk must be addressed in order to establish monotonicity of choice. Comparative static analysis is carried out with respect to changes in income, changes in uncertainty and changes in preferences and it is shown that the appropriate conditions for comparative statics are superextremal and single crossing property conditions with respect to appropriate value lattices (Antoniadou 1996). These are extensions of revealed preference conditions and in the case of a take-it-or-leave-it lottery choice, they are both necessary and sufficient. Furthermore, these conditions are implied by the standard conditions based on the Arrow-Pratt measure of absolute risk aversion in the univariate case. The case of Leontief preferences is studies extensively to demonstrate the work.