Unification of models for choice between delayed reinforcers.
Two famous choice models collapse into one equation, so pick the simpler form for your graphs.
01Research in Context
What this study did
King et al. (1990) took two rival math models of choice. One was delay-reduction theory. The other was amended incentive theory. Both try to explain why animals pick one delayed reward over another.
The team proved the two models boil down to the same equation. Once rewritten, they make identical predictions. No lab test can tell them apart.
What they found
The two theories are not rivals. They are two labels for one formula. Any data that fit one will fit the other. Researchers must look elsewhere to pick a winner.
How this fits with other research
Macdonald et al. (1973) showed pigeons match response ratios to reward ratios. That fact is what both models try to explain. King et al. (1990) reveal the explanations are the same math in different clothes.
Davison et al. (1989) found pigeons ignore overall reinforcer-rate feedback. Their data set a boundary any unified model must meet. The new single equation passes that test.
Hall (2005) later showed standard matching fails when earning rates differ across keys. The unified delay model must absorb that tweak. The merger is a base, not a finish line.
Why it matters
When you run concurrent-schedule probes, stop hunting for the “correct” model. Pick the form that is easiest to graph or teach. Both roads lead to the same curve. Save your energy for testing new variables like reinforcer immediacy or response effort.
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Join Free →Plot your client's concurrent-schedule data with the delay-reduction form; if it fits, you're done—no need to test the incentive version.
02At a glance
03Original abstract
Two models for choice between delayed reinforcers, Fantino's delay-reduction theory and Killeen's incentive theory, are reviewed. Incentive theory is amended to incorporate the effects of arousal on alternate types of behavior that might block the reinforcement of the target behavior. This amended version is shown to differ from the delay-reduction theory in a term that is an exponential in incentive theory and a difference in delay-reduction theory. A power series approximation to the exponential generates a model that is formally identical with delay-reduction theory. Correlations between delay-reduction theory and the amended incentive theory show excellent congruence over a range of experimental conditions. Although the assumptions that gave rise to delay-reduction theory and incentive theory remain different and testable, the models deriving from the theories are unlikely to be discriminable by parametric experimental tests. This congruence of the models is recognized by naming the common model the delayed reinforcement model, which is then compared with other models of choice such as Killeen and Fetterman's (1988) behavioral theory of timing, Mazur's (1984) equivalence rule, and Vaughan's (1985) melioration theory.
Journal of the experimental analysis of behavior, 1990 · doi:10.1901/jeab.1990.53-189