At the start
8 stamps left to the reward
Close to the goal
2 stamps left to the reward
Evidence
The famous paper never ran a single rat. Hull (1932) formulated the hypothesis in a theoretical paper with no experiment of its own and derived its logarithmic shape from someone else’s maze data. That animals run faster closer to the goal is a prediction there, not an observation; Hull only tested it in 1934, with a straight runway.
In humans it was first shown by Kivetz, Urminsky and Zheng (2006). Across 949 coffee stamp cards the interval between purchases fell by 20 percent from the first stamp to the last. Cards that could not be redeemed showed the opposite pattern — a control that rules out habituation as the explanation.
Limits
The field findings come from a campus café with no competition and a music-rating portal; the main analysis uses redeemed cards only. For SaaS onboarding and checkouts the transfer is plausible but not directly supported. A slow-starting progress bar in web surveys markedly increased drop-offs; bars in general did not reliably reduce abandonment (Villar et al., 2013).