Response rate and changeover performance on concurrent variable-interval schedules.

Pigeons pecked two keys. Each key paid off on its own variable-interval timer. The birds could hop back and forth at any time. W et al. recorded every peck and switch. They wanted math that could predict how often the birds would jump keys.

Three short equations did the job. One equation used response rate. One used time on each key. One used obtained reinforcers. Together they predicted changeover rate almost perfectly. The data still showed the usual slight under-matching, but the fit was tight.

CATANIA (1963) and Macdonald et al. (1973) had already proved the matching law on concurrent VI schedules. W et al. kept the same pigeon setup but zoomed in on the switches themselves. Katz et al. (2003) later moved the reinforcer ratios every few seconds; choice still tracked, showing the 1978 equations hold even when the program wiggles.

Bell (1999) shrank the model to just two variables: time since last peck and last key picked. That sounds simpler, yet W et al.’s three-equation set stays useful because it keeps separate terms for responses, time, and reinforcers. You can pick the grain of analysis you need.

Hall (2005) uncovered a trap: if the two keys differ in how fast reinforcers are actually earned, plain matching fails. The 1978 equations assumed equal accessibility, so when earning rates split, you must swap in S’s expanded formula. The papers do not clash; they just flag when to add the extra term.

If you run concurrent schedules in a classroom or clinic, these equations give you a quick check. Count responses, time, and reinforcers, then see if the client’s switching matches the prediction. A big gap tells you extra variables—like earned versus obtained reinforcers—are in play, and you can adjust the program or add changeover delays on the fly.