The Price of Uncertainty in Present-Biased Planning

The tendency to overestimate immediate utility is a common cognitive bias. As a result people behave inconsistently over time and fail to reach long-term goals. Behavioral economics tries to help affected individuals by implementing external incentives. However, designing robust incentives is often difficult due to imperfect knowledge of the parameter $\beta \in (0,1]$ quantifying a person's present bias. Using the graphical model of Kleinberg and Oren, we approach this problem from an algorithmic perspective. Based on the assumption that the only information about $\beta$ is its membership in some set $B \subset (0,1]$, we distinguish between two models of uncertainty: one in which $\beta$ is fixed and one in which it varies over time. As our main result we show that the conceptual loss of efficiency incurred by incentives in the form of penalty fees is at most $2$ in the former and $1 + \max B/\min B$ in the latter model. We also give asymptotically matching lower bounds and approximation algorithms.