1. Chung and Herrnstein (1962) first proposed that relative response strength matched relative immediacy of reinforcement in simple concurrent schedules (“immediacy” being defined as the reciprocal of delay). For early models of hyperbolic delay discounting, see Ainslie (1974, 1975), Rachlin (1970), Rachlin and Green (1972) and Herrnstein (1981).
2. The main focus of Killeen's (2009) article is financial discounting; his use of “utility” corresponds to our use of “value”, and his use of (monetary) “value” to our use of “quantity”. Thus, in Killeen (2009), marginal utility is expressed as du/dv, which corresponds, in the present article, to dV/dq.
3. The magnitude, or amount, effect is a ubiquitous finding in studies of inter-temporal choice in humans (Green et al., 1997; Kirby, 1997), but is not generally found with animals (Green et al., 2004; Vanderveldt et al., 2016). Rachlin, Arfer, Safin, and Yen (2015) have recently proposed a model which enables the magnitude effect to be accommodated by a modified hyperbolic model. Value is assumed to be a power function of reinforcer size, but, unlike other models discussed above, the exponent of the function varies as a function of delay: Vi ∝ qm, Vd ∝qn (n and m are generally < 1). The size of an immediate reward (qi) whose value equates to that of a larger delayed reward (qd) is given by qi = qd n/m/(1+K·d)n/m. The nature of the relation between the delay and the exponent remains to be determined.
4. Null equations have proved invaluable in other branches of biological science, most notably in classical pharmacology, where interpretation of the parameters of empirical dose-response relations has historically been hampered by elusive effector mechanisms intervening between receptor occupancy and measurable biological response (Arunlakshana and Schild, 1959; Elhert, 1988; see Kenakin, 1997 for a review).
5. The theoretical approach adopted by Valencia-Torres et al. (2011) differs somewhat from that of Marshall, Smith, and Kirkpatrick (2014), Kirkpatrick, Marshall, and Smith (2015), and Marshall and Kirkpatrick (2016). These authors have shown that selection of the larger, more delayed reinforcer in an inter-temporal choice task can be enhanced by exposure to temporal and magnitude training outside the inter-temporal choice context. They imply that delay and size are separate discriminanda, in contrast to the model of Valencia-Torres et al. (2011), which treats value as a composite discriminandum. To date, there has been no attempt to compare the two approaches empirically. For a detailed treatment of learning principles as applied to the neurobiology of inter-temporal choice, see Schultz (2015).