Discontinuity and Nonmonotonicity of Behaviors
in the Degree of Partial Naivete

This paper studies an implication of partial naivete in the environment studied in O'Donoghue and Rabin (1999): an agent, who is partially aware of her present bias, decides when to begin a single task that takes one period for completion within a finite number of periods. We show that an agent's decision changes discontinuously and non-monotonically in her sophistication level (or the degree of her partial naivete). There are at most two thresholds in the sophistication level: above the higher one, the partial naif behaves exactly like pure naifs, and below the lower one, the partial naif behaves exactly like pure sophisticates. This implies that the same conclusion in O'Donoghue and Rabin (2001) is robust even when the cost and reward schedules are arbitrary in a finite horizon. Moreover, we provide an example in which an agent completes the task at the earliest date when her sophistication level lies in the intermediate range between the two thresholds.