Hyperbolic discounting (Ainslie; Mazur)

V = A / (1 + kD)

• V = present subjective value · A = the reward's undiscounted amount · D = delay · k = the individual's discount rate. Value falls hyperbolically with delay — not exponentially.

Ratio form, and the hyperbola is the whole point. Because the curve is hyperbolic (steep at first, then flattening), preferences reverse: a larger-later reward preferred from a distance loses to a smaller-sooner one as the sooner reward approaches (the crossing of two hyperbolas). This is the mathematics of "I'll start tomorrow," impulse, and present bias.

Transfer. Immediacy is a multiplier on perceived value, and a distant payoff is discounted hard — the reason a hook must promise a soon payoff (hormozi-value-equation's time-delay term made precise), and the reason CTAs work better framed as an immediate small action than a large future benefit (fogg-behavior-model: make the near step easy). Also why "limited time" scarcity moves behavior — it collapses D.

Kin: illustrative-formulas · hormozi-value-equation · fogg-behavior-model · two-pleasures-arousal-and-closure (来劲 is present-tense by construction).

Source: George Ainslie, "Specious Reward" (Psychological Bulletin, 1975); functional form V = A/(1+kD) standardized by James Mazur (1987)