Making Decisions Using Sets of Probabilities: Updating, Time Consistency, and Calibration
This work addresses foundational issues in decision theory under ambiguity for researchers in economics and AI, though it is incremental as it builds on existing minimax and rectangularity frameworks.
The paper tackles the problem of how an agent should update beliefs represented by a set of probability distributions when using the minimax criterion, showing that anomalies like time inconsistency arise from different game-theoretic scenarios against a bookie with varying information, and characterizes cases where optimal rules involve conditioning or ignoring information.
We consider how an agent should update her beliefs when her beliefs are represented by a set P of probability distributions, given that the agent makes decisions using the minimax criterion, perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between updating and calibration when uncertainty is described by sets of probabilities. Our results emphasize the key role of the rectangularity condition of Epstein and Schneider.