Decision under ambiguity, composed optimization, and quantal response equilibria

Abstract

This paper looks at decision problems under ambiguity from a computational perspective. In common frameworks for handling ambiguity, an optimization objective is used that is constructed in two steps. In the first step, a vector-valued evaluation mapping is applied to a proposed decision. In the second step, the entries of the resulting vector are aggregated into a single number, which defines the objective value for optimization. In several standard decision models under ambiguity, one can define the evaluation mapping in such a way that optimization is relatively easy when the aggregation step is of linear type, i.e., a weighted sum. Even when the aggregator that the decision model actually prescribes is nonlinear, one may then still make use of a method known as iterative reweighting. In this method, the optimal decision is found by iteratively solving a series of weighted-sum problems. Our contributions are to provide a general treatment of this method in the framework of quasiconvex analysis, and to give conditions under which the fixed point can be viewed as representing a quantal response equilibrium in a population game, allowing the iterative reweighting method to be viewed as an evolutionary algorithm. In the context of ambiguity in beliefs, the fixed point can be interpreted as providing a set of ambiguity-neutral probabilities associated with the ambiguity states. As an illustration, we compute and compare the results of several different decision models in a portfolio optimization problem, both under ambiguity in beliefs and under ambiguity in preferences.

Keywords: Ambiguity, Composed Optimization, Quantal Response Equilibrium

JEL Classification: D81, C73, C63