Hyperbolic discount curves: a reply to Ainslie

Abstract

Ainslie (Theory and Decision, 73, 3-34, 2012) challenges our interpretation of the properties of hyperbolic discount curves in an iterated prisoners' dilemma (IPD) model. In this reply, we discuss the emergence of hyperbolic discount functions in the behavioral economics literature and evaluate their properties. Furthermore, we present a summarized version of our IPD model and evaluate Ainslie's points of contention.

The improving sequence effect on monetary sequences

“…As stated by Musau (2014) , , that is, is strictly monotone decreasing. Moreover, Equation (4) shows that the first derivative of the function with respect to k is negative: …”

“…The one-parameter model by Mazur (1984) is the simplest and the most common representation of hyperbolic discounting. This model is used to explain inconsistent behavior of individuals, since it is experimentally observed that the discount rate is not constant over time, as the discounted utility model requires ( Frederick et al, 2002 , Musau, 2014 ).…”

A Mathematical Analysis of the Improving Sequence Effect for Monetary Rewards

“…Proposition 2 shows that, in order to explain the improving sequence effect, it is not possible to consider a separable discount function [17]. So, our first task was to define a non-separable discount function.…”

Measuring Impatience in Intertemporal Choice

“…Nevertheless, and as indicated at the beginning of this Section, this discounting model can be simplified by using a function F ( t ) independent of delay d . More specifically, a one-variable discount function F ( t ) ([ 38 ] and [ 39 ]) is a continuous real function such that defined within an interval [0, t 0 ) ( t 0 can even be +∞), where F ( t ) represents the value at 0 of a $1 reward available at instant t , satisfying the following conditions: F (0) = 1, F ( t ) > 0, and F ( t ) is strictly decreasing. …”