A Game-Theoretic Analysis of Updating Sets of Probabilities (original) (raw)

Abstract We consider how an agent should update her un- certainty 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 mini- max 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 reason- able 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 dif- ferent information. We characterize the impor- tant special cases in which the optimal decision rules according to the minimax,criterion amount to either conditioning or simply ignoring the in- formation. Finally, we consider the relationship between conditioning and calibration ...