The case against Allen's generalization of the matching law.

Houston (1982) wrote a math check on Allen’s power-function version of the matching law.

The paper shows the formula breaks when two schedules are paired or shown at the same time.

No kids, no pigeons—just equations and logic.

The proof says Allen’s form cannot describe choice when reinforcers overlap.

In plain words, the pretty curve does not fit real concurrent schedules.

Mellitz et al. (1983) stepped in next year. They offered a hill-climbing rule that lets momentary maximizing create matching without a separate law. Their model bypasses the algebra problem Houston (1982) exposed.

Hall (1992) tested kids in a classroom token system. Matching appeared only when reinforcer quality was equal. That boundary echoes I’s warning that the simple equation needs extra terms.

Schenk et al. (2020) and Johnson et al. (2009) both found the generalized matching equation fits basketball shot data. These field studies seem to contradict I’s critique, but they use undermatching and bias parameters—exactly the looseness I said Allen’s tight power form lacks.

When you plot concurrent-schedule data, do not trust a single power curve. Check for undermatching, bias, and quality differences. If the numbers drift, add parameters or switch to a maximizing model like Mellitz et al. (1983). This keeps your assessment honest and your treatment recommendations solid.