Towards a Purely Behavioral Definition of Loss Aversion

Abstract

This paper suggests a behavioural, preference-based definition of loss aversion for decision under risk. This definition is based on the initial intuition of Markowitz, and Kahneman and Tversky that most individuals dislike symmetric bets, and that the aversion to such bets increases with the size of the stake. A natural interpretation of this intuition leads to defining loss aversion as a particular kind of risk aversion. The notions of weak loss aversion and strong loss aversion are introduced, by analogy to the notions of weak and strong risk aversion. We then show how the proposed definitions naturally extend those of Kahneman and Tversky, Schmidt and Zank, and Zank from a setting of lotteries to a setting of arbitrary (continuous) distributions, which is helpful for applications. The implications of these definitions under Cumulative Prospect Theory (PT) and Expected-Utility Theory (EUT) are examined. In particular, we show that in EUT loss aversion is not equivalent to the utility function having an S shape: loss aversion in EUT holds for a class of utility functions that includes S-shaped functions, but is strictly larger than the collection of these functions. This class also includes utility functions that are of the Friedman-Savage type over both gains and losses, and utility functions such as the one postulated by Markowitz. Finally, we discuss possible ways in which one can define an index of loss aversion for preferences that satisfy certain conditions. These conditions are satisfied by preferences having a PT-representation or an EUT-representation. Under PT, the proposed index is shown to coincide with Kobberling and Wakker’s index of loss aversion only when the probability weights for gains and losses are equal. In Appendix B, we consider some extensions of the study done in this paper, one of which is an extension to situations of decision under uncertainty with probabilistically sophisticated preferences, in the sense of Machina and Schmeidler.