The arithmetic of discounting.

Killeen (2015) wrote new math rules for delay discounting.

The paper stays on the whiteboard—no kids, no pigeons, no data.

It asks: what if discounting adds and subtracts instead of multiplying?

The model shows that small versus large rewards can change how steeply value drops.

Magnitude effects pop out naturally when you use plus and minus, not times and divide.

Killeen (2020) is the next chapter. It keeps the same goal—better equations—but swaps the adding idea for a new spending function.

Oliveira et al. (2014) tested real pigeons with concurrent-chains and still saw the classic hyperbolic curve. Their data fit the old multiply style, yet the birds don’t contradict Killeen (2015); they simply weren’t arranged to show additive parts.

Cullinan et al. (2001) and Grosch et al. (1981) show that a simple signal during the delay lifts preference. Those facts feed both models: they tell you what to put into the formulas, not which math sign to use.

When you run a discounting assessment, plot the data both ways. If the standard hyperbolic line looks off, try adding utility and disutility instead of multiplying them. The move costs nothing—just a different Excel column—and may reveal why some clients seem to over-value big delayed rewards.