Formal properties of the matching law.

Michael (1974) wrote the math behind the matching law. No birds, no kids, just equations.

The paper spells out how response ratios lock onto reinforcement ratios. It skips size of reward and hunger level.

The core rule: behavior splits the same way reinforcement splits. Magnitude and drive do not touch the final split.

The math holds once responding settles. Early swings don’t count.

Allen (1981) and Marr (1989) added power-function forms. They show the 1974 straight-line rule bends when reinforcers are lopsided. The new math keeps the spirit but lets you fit real-world curves.

Oliver et al. (2002) took the same rule into homes with severe problem behavior. Caregiver attention ratios predicted bite ratios. Lab math worked at the kitchen table.

Hastings et al. (2001) later showed pigeon data keep the same slope even when seed size changes. This backs J’s claim that magnitude is ignored once ratios stabilize.

You now have a pocket calculator for choice. Count the reinforcers on each option, not their size, and you can forecast how behavior will settle. If attention for screaming is 80% of all attention, expect 80% of responses to be screams. Shift the ratio, wait for steady state, and the numbers will follow. Use power-function forms when reinforcers are uneven, like screen time versus grapes.