A distance-based theory of lottery complexity

Abstract

This paper proposes a metric approach to measuring the complexity of lotteries. Starting by observing that degenerate lotteries are the simplest choice alternatives, the complexity of a lottery is evaluated by its distance from the closest degenerate lottery. Equivalently, a lottery is complex when it is difficult to approximate it by a single outcome. Given a metric over outcomes, the complexity index we consider is the minimum average distance between the lottery and one of its best degenerate proxies. The paper provides an axiomatic foundation for this representation, studies its main properties, and compares it with other measures of complexity. It then applies the index to choice under risk through a class of complexity adjusted expected utility preferences. Within this model, we study how complexity affects risk attitudes and consistency with stochastic dominance.