Delay of reinforcers in a concurrent-chain schedule: An extension of the hyperbolic-decay model.
Reinforcer value drops along a hyperbolic curve, and longer entry waits make both choices lose pull faster.
01Research in Context
What this study did
Pigeons pecked two keys in a two-part chain. First part was short. Second part had different wait times to food.
The author changed how long the first part lasted and how long the birds waited in the second part. He checked if a simple hyperbolic curve still predicted their choices.
What they found
The hyperbolic decay rule still fit. As the wait to the food links grew, the birds’ choices flattened out.
Longer first-part waits made both options lose value faster, but the math still worked.
How this fits with other research
Delano (2007) extends this idea. Pigeons faced mixed or fixed work schedules before the same delay. The curve still held, showing schedule type can shift choice even when delay and payoff stay put.
Lloyd (2002) seems to contradict. Keeping a 10-second gap between delays did not keep choice constant. The birds cared about total wait, not just the gap. The 1988 model already lets total wait matter, so the papers actually agree once you read the fine print.
Haemmerlie (1983) came first. It showed bigger-later food loses appeal as delays grow. The 1988 paper keeps that core and adds the hyperbolic math to cover any first-part length.
Why it matters
You now have a quick rule: value = reward ÷ (1 + k × delay). Use it when you set up token boards, chained schedules, or DRO plans. If the wait to the big reinforcer is long, add an early small win or shorten the chain. The curve tells you how much the big prize fades, so you can adjust before the client drifts away.
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02At a glance
03Original abstract
Six pigeons were trained in concurrent-chain schedules with equal aperiodic initial links and delays to reinforcers in the terminal links. The terminal links always lasted 30 s. In Experiment 1, two reinforcers were delivered in each terminal link, with the first reinforcer delivered either 1 s (Experiment 1A) or 5 s (Experiment 1B) after choice. In these experiments, the delay between the first and second reinforcers in one terminal link was 10 s, and the delay between the first and second reinforcers on the other key was varied. This variation produced little change in preference. In Experiment 1C, the first and second delays on one key were 10 s, and on the other key they were varied within the restriction that the sum of delays was 20 s. Preference for the varied terminal link increased as the first delay was decreased. A hyperbolic model of the value of reinforcer delay provided a good description of the data from Experiment 1. In Experiment 2, a single reinforcer was delivered in each terminal link after a delay of either 0.2 or 19.8 s, and these delays were reversed between conditions. The initial-link schedule providing terminal-link access was varied from means of 5 s to 480 s. As the initial-link duration was increased, preference for the shorter delay became less extreme. An extension of the hyperbolic-decay model, in which the decay constant was a hyperbolic function of the initial-link duration, described the results well. Differences between the procedure used here (constant-duration terminal links) and that used in conventional concurrent-chain research precludes use of the model as a general account of concurrent-chain performance.
Journal of the experimental analysis of behavior, 1988 · doi:10.1901/jeab.1988.50-219