On the measurement of reinforcement frequency in the study of preference.

Shimp (1968) tested how pigeons choose between two keys. Each key gave food on its own variable schedule. The birds could switch keys at any time.

The team compared two ways to average the payoff rates. One way adds and divides. The other way uses the harmonic mean. They asked which math the birds followed.

The pigeons' key preference lined up with the harmonic mean, not the simple average. When the schedules changed, the birds' choices shifted the same way the harmonic prediction did.

This means the birds acted as if they 'felt' the true density of payoffs, not just a raw count.

HERRNSTEIN (1961) had already shown that pigeons match their pecks almost perfectly to minimum response requirements. Shimp (1968) keeps that finding but adds the finer point: the math you use to compute 'frequency' matters.

Elsmore et al. (1994) later moved the same idea into an eight-arm maze with rats. Again, nose-poke rates tracked reinforcement best when the harmonic view was used, showing the rule crosses species and setups.

Mandell (1984) flipped the question sideways: instead of using rates to predict choice, he asked if pigeons can tell frequencies apart when the events are just signals. They can, but control fades with delay. Together these papers show both sides—organisms use, and also notice, event density.

When you write a concurrent-schedule program or graph a matching analysis, average your reinforcers with the harmonic mean. The birds—and probably your clients—already do. Using the simple average can make data look like a mismatch when it is not. Switch the formula and your predictions, and your treatment decisions, get sharper.