Optimization and the matching law as accounts of instrumental behavior.

Allen (1981) compared two big ideas about why we do what we do. One idea says we match our behavior to the pay-off rates. The other says we always pick the richest single option.

The paper looked at which idea fits best under different lab schedules. It asked: does matching or maximizing better explain pigeons pressing levers for food?

Matching won for concurrent schedules. When two choices run at the same time, birds split their responses to match the payoff rates.

Maximizing won for single schedules. When only one option is open, birds press at the rate that pulls the most food.

Mates (1990) later showed the fight was a false alarm. Under concurrent-chains, the math of matching and maximizing can collapse into the same formula.

Avellaneda et al. (2025) now supersedes the old rule. Their new matching law lets sensitivity drift with overall rate, giving tighter fits than the 1981 version.

Hoch et al. (2007) extends the idea to people. They tracked how adults with developmental disabilities split their problem behavior versus polite bids for staff attention. The same matching equation fit the real-world data.

You now have a quick decision tree. Use matching when you run concurrent reinforcement like choice sessions or competing schedules of attention. Use maximizing when you shape a single skill and want the highest rate of reinforcement possible. Keep changeover delays short or matching will flatten. And if the data look off, try the newer rate-sensitive matching form instead of forcing the 1981 classic.