An analytic comparison of Herrnstein's equations and a multivariate rate equation.

Hammond (1980) compared two math models that predict how fast animals respond on VI schedules.

The first model was Herrnstein's famous hyperbola. The second was a newer linear-system rate equation.

Using algebra, J showed the newer equation lets the top rate k change when reward size or timing changes.

The math proved k is not a fixed ceiling. It slides up or down with how rich or lean the schedule is.

This means the classic hyperbola can mislead you if you treat k as constant across conditions.

Webb et al. (1999) ran thirsty pigeons and watched k drop when water deprivation eased. Their data matched Hammond (1980)'s math warning that k moves.

Wright (1972) had earlier shown the hyperbola works for avoidance, but that study assumed k stayed put. Hammond (1980) adds a caution: the same equation may need a sliding k when reward factors shift.

Kessel et al. (2008) later built another closed-form model for IRT patterns. Both papers share a goal: replace rough rules with exact equations that fit real data.

When you graph VI data using Herrnstein's hyperbola, let k float instead of locking it. Fit separate k values for each reward size or delay you test. This simple tweak gives cleaner curves and better predictions in your next session.