A Game-Theoretic Analysis of Updating Sets of Probabilities
This work addresses foundational issues in decision theory for agents using minimax criteria, but it is incremental as it builds on existing game-theoretic analyses of uncertainty.
The paper tackles the problem of how an agent should update uncertainty represented by a set of probability distributions upon observing new data, using a game-theoretic framework with a bookie. It shows that anomalies like time inconsistency arise from different informational games and characterizes when optimal decisions involve conditioning or ignoring information.
We consider how an agent should update her uncertainty when it is represented by a set P of probability distributions and the agent observes that a random variable X takes on value x, 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 conditioning and calibration when uncertainty is described by sets of probabilities.