The maximization of overall reinforcement rate on concurrent chains.

Houston et al. (1987) built a math model of concurrent-chain schedules. They asked: do animals pick the chain that gives the most total reward per minute?

The model kept reinforcement rate constant but let reward size vary across chains. It predicted general trends, not exact choice splits.

When one chain delivered bigger pay-offs, animals moved toward it, but not as far as the rate-maximizing formula said they should.

The paper labels this a systematic deviation: choice follows the direction of maximization yet undershoots the precise optimal split.

Fovel et al. (1989) later showed mice doing almost perfect maximizing on VR-VI chains when reward duration, not size, was varied. The two studies sit side-by-side: size differences create deviation; duration differences do not.

Hinson (1988) added another wrinkle. He varied both rate and duration at once and saw animals slide toward indifference as overall rate rose. That empirical result sharpens the 1987 model by showing rate changes can wash out sensitivity to other dimensions.

Yuwiler et al. (1992) looked like a direct contradiction: pigeons on a VI analogue matched rather than maximized even when maximization should win. The clash fades once you note the species and the task structure differ; the core message stays—simple rate math cannot stand alone.

If you run concurrent schedules in practice, remember that bigger reinforcers pull behavior only part-way toward the mathematical best. You will need extra prompts or time to reach optimal allocation. Also, watch for duration or rate shifts elsewhere in the setting—they can override the size effect and push clients toward indifference.