Discounting of delayed rewards is not hyperbolic

Abstract

Delay discounting refers to decision-makers' tendency to value immediately available goods more than identical goods available only after some delay. In violation of standard economic theory, decisionmakers frequently exhibit dynamic inconsistency; their preferences change simply due to the passage of time. The standard explanation for this behavior has appealed to the nature of decision-makers' discount functions, specifically positing a hyperbolic discount function. Though this explanation has been largely accepted, there has been surprisingly little work examining whether preference reversals are actually consistent with hyperbolic discounting. The current study holds hyperbolic discounting to the same empirical standard that exponential discounting has been held to and finds that choice behavior is not consistent with hyperbolic discounting. Despite the overwhelming focus placed on hyperbolic discounting, the current findings cast doubt on hyperbolic discounting as an explanation of decision-makers' undesirable preference reversals and as an explanation of delay discounting behavior in general. Keywords: delay discounting, intertemporal choice, decision making, dynamic inconsistency, hyperbolic Delay discounting refers to the robust finding that animals, including humans, behave as though immediately consumable goods are more valuable than those only available after some delay. For example, a decision-maker might choose $100 delivered immediately over$200 to be delivered in 3 years. One of the major empirical questions regarding such choices has focused on the discount function-the mathematical function specifying the relationship between reward magnitude, delay, and subjective value. Initial theoretical work in economics where V D represents the current value of a reward that will be delivered after a delay D, k represents the decision-maker's discount rate, and V 0 represents the undiscounted value of that same reward (i.e., the value of that reward if it were available immediately). Unfortunately, the vast majority of empirical evidence has demonstrated that humans The distinction between exponential and hyperbolic discounting may, at first, appear to be a bit of mathematical nitpicking. However, these two different discounting schemes represent very different conceptualizations of how delayed rewards are evaluated and have serious implications for behavior. Exponential discounting represents how bank loans work; the value of a delayed reward declines by a fixed percentage per unit of time. Hyperbolic discounters, on the other hand, behave as though their discount rates increase as the delivery of a delayed reward draws near. Thus, hyperbolic discounters appear to exhibit patience when dealing with rewards in the distant future only to find that this patience diminishes as rewards move closer in time. The critical problem with the diminishing patience exhibited by hyperbolic discounters is that it tends to produce preferences that shift in systematic and predicable ways due to the simple passage of time; what economists refer to as dynamic inconsistency. For example, a hyperbolic discounter might prefer $200 delivered in 2 years over$100 delivered in 1 year (a patient preference) only to reverse this preference a year later, preferring an immediate $100 over$200 delivered in 1 year (an impatient preference). 1 Because the former pair of rewards will become the latter pair of rewards in 1 year, the preferences described by hyperbolic discounting are 1 To be clear, hyperbolic discounting does not imply that preferences over an arbitrary pair of rewards will reverse for any change in delay. Whether or not a hyperbolic discounter reverses their preferences as time passes depends on the specific reward magnitudes, delays, and discount rates. That being said, changes in delay always alter the strength of a hyperbolic discounter's preference (i.e., the difference between the discounted value of the two rewards) in the predicted direction, even when actual choices do not reverse. In addition, reward pairs and an appropriate delay can always be found such that any non-exponential discounter (hyperbolic or otherwise) is guaranteed to reverse their preference (and thus represent an arbitrage opportunity)