Quasi-dynamic choice models: Melioration and ratio invariance.
Ratio invariance outruns melioration for two-choice probabilistic schedules, yet newer models now handle three options and moment-by-moment shifts.
01Research in Context
What this study did
Mazur (1988) compared two math rules that describe how animals choose between two levers.
One rule is called melioration. It says the animal keeps switching until both levers feel equally good.
The other rule is ratio invariance. It says the animal locks onto a fixed ratio of responses no matter how the pay-offs bob around.
The paper used algebra, not live birds, to ask which rule survives when the schedule odds keep changing.
What they found
Ratio invariance won. It predicted the birds’ response splits better and avoided “behavioral traps” where melioration would waste time on lean schedules.
The model also stayed steady when the experimenters shifted reward probabilities, making it the safer bet for dynamic situations.
How this fits with other research
Zigler et al. (1989) ran computer birds through the same contest and got the same winner. Their simulation backs up the 1988 algebra, forming a clean replication pair.
Navakatikyan et al. (2013) later stepped beyond two levers. They showed that when three or more choices sit on the table, a newer “component-functions” model beats both old rules. The field has therefore moved past ratio invariance for complex cases.
Tyrer et al. (2009) looked at micro-preference pulses in real pigeons. They found short-lived swings that none of the static rules, including ratio invariance, fully catch. This hints that even the winning 1988 rule needs a time-based upgrade.
Why it matters
If you write concurrent-schedule programs or use matching-law probes, pick equations that stay stable when contingencies drift. Ratio invariance gives you that backbone, but switch to Alexander’s component model when you add a third option. Track moment-to-moment data if you want to catch the brief swings F et al. showed — simple averages will hide them.
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Join Free →Plot your client’s concurrent-schedule responses on a log ratio chart; if the slope stays flat while probabilities shift, ratio invariance is doing the work.
02At a glance
03Original abstract
There is continuing controversy about the behavioral process or processes that underlie the major regularities of free-operant choice such as molar matching and systematic deviations therefrom. A recent interchange between Vaughan and Silberberg and Ziriax concerned the relative merits of melioration, and a computer simulation of molecular maximizing. There are difficulties in evaluating theories expressed as computer programs because many arbitrary decisions must often be made in order to get the programs to operate. I therefore propose an alternative form of model that I term quasi-dynamic as a useful intermediate form of theory appropriate to our current state of knowledge about free-operant choice. Quasi-dynamic models resemble the game-theoretic analyses now commonplace in biology in that they can predict stable and unstable equilibria but not dynamic properties such as learning curves. It is possible to interpret melioration as a quasi-dynamic model. An alternative quasi-dynamic model for probabilistic choice, ratio invariance, has been proposed by Horner and Staddon. The present paper compares the predictions of melioration and ratio invariance for five experimental situations: concurrent variable-interval variable-interval schedules, concurrent variable-interval variable-ratio schedules, the two-armed bandit (concurrent random-ratio schedules), and two types of frequency-dependent schedule. Neither approach easily explains all the data, but ratio invariance seems to provide a better picture of pigeons' response to probabilistic choice procedures. Ratio invariance is also more adaptive (less susceptible to "traps") and closer to the original expression of the law of effect than pure hill-climbing processes such as momentary maximizing and melioration, although such processes may come in to play on more complex procedures that provide opportunities for temporal discrimination.
Journal of the experimental analysis of behavior, 1988 · doi:10.1901/jeab.1988.49-303