Pigeons' choices between fixed-ratio and linear or geometric escalating schedules.
The sums-of-reciprocals rule still predicts choice when the required responses rise linearly or geometrically.
01Research in Context
What this study did
Hagopian et al. (2000) let pigeons pick between two keys. One key always asked for the same number of pecks. The other key asked for more pecks each time, either by the same step or by the same percent.
The birds worked in a little box. Every peck counted. The team watched which key the bird chose and how many times it pecked.
What they found
The birds followed the sums-of-reciprocals rule. That rule says: add up 1 divided by each ratio, then pick the side with the bigger total.
It worked for both kinds of rising ratios. Linear steps or geometric steps, the rule still predicted the birds’ choices.
How this fits with other research
Shimp et al. (1971) first showed pigeons match their pecks to how often each side pays off. The new study keeps that idea, but tests it when the price keeps rising.
Hamm et al. (1978) proved matching works for both good and bad outcomes. Hagopian et al. (2000) now show it also works when the work load grows, not just when the payoff rate changes.
Parsons et al. (2013) saw birds shift choice slowly after each payoff. The 2000 birds shifted instantly, because each key showed its whole price up front. Same law, different speed.
Why it matters
If you use progressive ratio schedules with clients, you now know the simple math that guides choice. Tell the learner the whole price list up front, and their picking should follow the sums-of-reciprocals rule. You can plan breaks or rewards so the math stays on your side.
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02At a glance
03Original abstract
Four related procedures provided a basis for comparing the linear-optimality principle with a principle based on the sums of reciprocals of distances to reinforcement, and to explore the generality of the sums-of-reciprocals principle as a description of choice patterns in situations of diminishing returns. The procedures all arranged choices between fixed-ratio schedules and progressive-ratio schedules, which escalated with each consecutive choice. In contrast to previous work that involved constant ratio increments, two sets of procedures in this study involved relatively small increments that are similar to the early values when a progressive schedule is increasing proportionally. The remaining two sets of procedures examined progressive schedules with proportional increments. In addition, the initial value of the progressive alternative was manipulated to determine its effects on patterns of choice with both linear and proportional types of escalation. With the exception of one phase, regardless of the initial/reset value and the patterns of escalation, patterns of choice with pigeons were well characterized by the sums-of-reciprocals principle. This supports previous research with pigeons using fixed-increment progressive schedules, as well as situations in which the progressive schedule increased by constant proportions instead of by constant increments. The findings are attributed to the feature of this averaging technique whereby it differentially values reinforcers based on their relative proximity to a particular choice point.
Journal of the experimental analysis of behavior, 2000 · doi:10.1901/jeab.2000.73-93