Probabilistic models of delay discounting: “Fixed-endpoint” psychometric curves improve plausibility and performance

Probabilistic models of delay discounting allow the estimation of discount functions without prescribing unrealistically sharp boundaries in decision making. However, existing probabilistic models have two implausible implications: first, that no reward is sometimes preferred over some reward (e.g., $0 now over $100 in 1 year), and second, that the same reward is sometimes preferred later rather than sooner (e.g., $100 in a year over $100 now). We introduce a class of “fixed-endpoint” models that assign these edge cases a probability of 0. We find that these outperform conventional models across a range of discount functions using nonlinear regression. We also introduce a series of generalized linear models that implicitly parameterize various discount functions, and demonstrate the same result for these.