Web agents increasingly serve the same user across repeated online tasks such as booking travel, shopping, and scheduling, so their long-term usefulness depends on how well they know the user's preferences. Passive memory and personalization methods cannot obtain preferences the user never reveals, and clarification methods that value a question by its benefit to the current task overlook the benefit of reusing the answer in later tasks. We study Active Preference Acquisition (APA): which missing preferences are worth the cost of asking? We define the Prospective Personalization Value (PPV) of a missing preference as its expected cumulative utility gain over future tasks minus the effort and disclosure-risk cost of asking, and select questions under a per-interaction budget. We characterize when cumulative reuse justifies acquisition despite insufficient single-task benefit, and evaluate this regime with an initial-profile diagnostic. APA computes the counterfactual gain of each candidate preference on public task prototypes with the agent's own task solver and aggregates it by the expected reuse of each task family. We evaluate it on ActiveWebPref, a controlled longitudinal benchmark with hidden but queryable user profiles and exact task utilities. At a horizon of 20 tasks, 34.4% of budget-feasible valuable missing preferences are amortization-dependent; APA acquires 80.3% of them, whereas current-task value of information acquires 5.1%. Under per-user asking-cost caps matched to APA's realized expenditure, APA achieves higher net utility than four heuristic selection baselines. In our cost setting, asking about every unknown preference yields lower net utility than not asking, while APA recovers 84.2% of its utility gain with 27.6% of its questions. The value of acquiring a user preference should account for its reuse across future tasks, not only its benefit to the task at hand.

翻译：暂无翻译