Bidding for delayed rewards: Accumulation as delay discounting, delay discounting as regulation, demand functions as corollary

Abstract

In the accumulation paradigm animals press one manipulandum to accumulate pellets or seconds of access to food, and then press another manipulandum, or run some distance, to collect it. The accumulation may be interpreted as delay discounting, with the animals adjusting the distal amount to compensate for its distance or delay. The amount accumulated before being collected is a linear function of the distance or time that the experimental paradigm stipulates for collecting it. That linear function follows from the Unit‐Price/Unit‐Amount axiom. The inverse of the linear compensation functions gives a delay‐discount hyperbola. The advantages of the accumulation paradigm and analytic framework for delay discounting studies are noted. Compensation functions are then derived from a behavioral regulation model, which generalizes them to contexts where the individual's budget for response cost becomes over‐taxed. In turn, such compensation/regulation models lead directly to representative demand functions. In sum, regulation models provide a theoretical grounding for demand functions, compensation functions, and delay discounting hyperbolas. The parameter that links them is the unit amount k, the slope of the compensation function and of the discount function, the setpoint for consumption in a regulation model, and the ideal quantity consumed at minimal price in a demand analysis.

Addendum to Killeen's (2019) Bidding for Delayed Rewards

“…By this metric, not having a reusable bag was twice as important (relative to their costs) for the environmentally concerned as for the unconcerned group. As predicted by Killeen (2019), the minimum distance model can generate credible demand and expenditure functions. As the case with all nonlinear models, Equation 8b should be tested against the data of individuals, but that is not attempted here.…”

“…The longer the duration, the greater that a must be to justify its delay. In this case, the one intended in Killeen (2019), the bid is like contract labor: I will work for you for a day if you will pay me a . If the contract is for a week or a month, I will demand more; a will be greater.…”

“…The appendix of (Killeen, 2019) encouraged investigators to use the harmonic mean in averaging data both within and across subjects but lacked a formal justification for that recommendation. A justification is provided by Myung et al (2000), but for the use of a geometric mean.…”

“…Killeen (2019) used Staddon's regulation model to extend the compensation function to the case of large costs and benefits, where the unit amount axiom can fail. The accumulation paradigm permits the animal to increase the benefit to compensate for the imposed increase in costs, rather than, as done in demand analyses, chart the decreased consumption with fixed benefits.…”

“…The current note is an attempt to remedy shortcomings of the recent article “Bidding for delayed rewards…” Killeen (2019), such as: Confusing terminology. Absence of a formal treatment of the most common paradigm in delay discounting: choice between goods of small value delivered at a short delay, and goods of greater value delivered at a longer delay. Failure to cite predecessors of that analysis. Demonstration of the phenomenological adequacy of his expenditure function. …”