Choice: Some quantitative relations.

E and colleagues ran pigeons in a two-key chamber. The birds first pecked on either key to enter a terminal link.

Only one terminal link ended in grain. The team varied how soon that grain arrived. They recorded which key the bird picked.

A plain delay-reduction equation missed some choices. Taking the square root of each reinforcement rate made the numbers line up better.

The tweak let one formula predict most of the birds' picks across many delay values.

Quilitch et al. (1973) had already squared delay values to fit simple concurrent schedules. Sarber et al. (1983) did the opposite—square-rooting rates—because their procedure added the extra 'chain' step. The two changes look opposite but serve different set-ups.

Jensen et al. (1973) squeezed 94% of variance out of pigeon data with a terminal-link model. The 1983 square-root move is a second, simpler way to reach similar accuracy under the same chained layout.

Later work kept borrowing the idea. Green et al. (1987) used delay-reduction to explain why birds avoid unreliable pay-offs. Hinson (1988) swapped the square root for a hyperbolic decay curve. Each paper tweaks the math, but all stay inside the delay-reduction family started here.

If you write token boards, chained schedules, or differential reinforcement plans, remember that raw rate alone can mislead. A quick square-root adjustment gives a closer guess of which side a client will choose. Try computing √rate for each branch the next time you set up a choice program, then watch if the prediction matches live data. The small math step costs nothing and can save you from accidental extinction of the 'losing' option.