Effects of adding a second reinforcement alternative: implications for Herrnstein's interpretation of r(e).
Adding a second VI schedule can make reinforcer rates covary and wreck Herrnstein’s constant-r(e) assumption.
01Research in Context
What this study did
Baker et al. (2005) tested adults in a small lab. Each person worked on two VI schedules at once.
The team added a second schedule and watched reinforcer rates on both. They wanted to see if the extra schedule stayed truly separate.
What they found
Reinforcer rates on the two schedules moved together. This breaks Herrnstein’s rule that outside reinforcement stays fixed.
When r(e) is not constant, the pretty hyperbola curve bends the wrong way.
How this fits with other research
Pear et al. (1984) showed that bigger payoffs slide the whole hyperbola up. Their r(e) was held steady, so the math still worked.
Glenn (1988) ran concurrent VR schedules and saw matching hold only when both counters tick together. Like L et al., extra schedules changed the payoff picture.
Oliver et al. (2002) mixed rate and magnitude and kept terminal links timed. They still treated reinforcer rates as independent, something L et al. warn may fail once a second VI joins.
Why it matters
If you run concurrent schedules in a token board or choice setup, remember that adding another option can let reinforcer rates bleed together. Check whether “outside” reinforcement really stays flat before you plug numbers into Herrnstein’s formula. When in doubt, measure both schedules, not just the one you care about.
Want CEUs on This Topic?
The ABA Clubhouse has 60+ free CEUs — live every Wednesday. Ethics, supervision & clinical topics.
Join Free →Before you call one schedule the ‘background,’ count reinforcers on both levers for ten minutes to be sure they don’t rise together.
02At a glance
03Original abstract
Herrnstein's hyperbola describes the relation between response rate and reinforcer rate on variable-interval (VI) schedules. According to Herrnstein's (1970) interpretation, the parameter r(e) represents the reinforcer rate extraneous to the alternative to which the equation is fitted (the target alternative). The hyperbola is based on an assumption that extraneous reinforcer rate remains constant with changes in reinforcer rate on the target alternative (the constant-r(e) assumption) and that matching with no bias and perfect sensitivity occurs between response and reinforcer ratios. In the present experiment, 12 rats pressed levers for food on a series of 10 VI schedules arranged on the target alternative. Across conditions, six VI values and extinction were arranged on a second alternative. Reinforcer rate on the second alternative, r2, negatively covaried with reinforcer rate on the target alternative for five of the six VI values on the second alternative, and significant degrees of bias and undermatching occurred in response ratios. Given covariation of reinforcer rate on the second and target alternatives, the constant-r(e) assumption can be maintained only by assuming that reinforcer rate from unmeasured background sources, rb, covaries with reinforcer rate on the second alternative such that their sum, r(e), remains constant. In a single-schedule arrangement, however, r(e) equals rb and thus rb is assumed to remain constant, forcing a conceptual inconsistency between single- and concurrent-schedule arrangements. Furthermore, although an alternative formulation of the hyperbola can account for variations in bias and sensitivity, the modified equation also is based on the constant-r(e) assumption and therefore suffers from the same logical problem as the hyperbola when reinforcer rate on the second alternative covaries with reinforcer rate on the target alternative.
Journal of the experimental analysis of behavior, 2005 · doi:10.1901/jeab.2005.09-05