Matching, delay-reduction, and maximizing models for choice in concurrent-chains schedules.
Under concurrent chains, matching, delay-reduction, and maximizing are the same equation wearing different hats.
01Research in Context
What this study did
Mates (1990) wrote math proofs instead of running animals. He asked: if you line up two schedules in a row (concurrent chains), do matching, delay-reduction, and maximizing models really fight each other? He showed that, with one extra assumption (power feedback), the three camps give the same numbers.
What they found
The paper found peace. When reinforcer size grows as a power of response rate, melioration and optimization predict the same choice curve. The models only look different because earlier papers used different math shortcuts.
How this fits with other research
Allen (1981) said matching beats optimization on concurrent schedules. Mates (1990) answers: the war ends if you add power feedback. The two papers talk past each other—M looked at simple concurrent, E looked at chained schedules.
Dougherty et al. (1996) later showed pigeons really do track delay differences, not ratios. That lab data gives the delay-reduction leg of E’s triangle a live test.
Avellaneda et al. (2025) updates the matching law itself, letting sensitivity drift with overall rate. Their tweak lives inside the same math umbrella E (199) proved equivalent to maximizing.
Why it matters
You can stop arguing with colleagues about which model is “true.” Pick the one whose parameters are easiest to measure in your setting. If you program concurrent-chain reinforcement, power feedback is now your free pass to use matching, delay-reduction, or maximizing formulas interchangeably.
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02At a glance
03Original abstract
Models of choice in concurrent-chains schedules are derived from melioration, generalized matching, and optimization. The resulting models are compared with those based on Fantino's (1969, 1981) delay-reduction hypothesis. It is found that all models involve the delay reduction factors (T - t2L) and (T - t2R), where T is the expected time to primary reinforcement and t2L, t2R are the durations of the terminal links. In particular, in the case of equal initial links, the model derived from melioration coincides with Fantino's original model for full (reliable) reinforcement and with the model proposed by Spetch and Dunn (1987) for percentage (unreliable) reinforcement. In the general case of unequal initial links, the model derived from melioration differs from the revised model advanced by Squires and Fantino (1971) only in the factors affecting the delay-reduction terms (T - t2L) and (T - t2R). The models of choice obtained by minimizing the expected time to reinforcement depend on the type of feedback functions used. In particular, if power feedback functions are used, the optimization model coincides with that obtained from melioration.
Journal of the experimental analysis of behavior, 1990 · doi:10.1901/jeab.1990.54-53