  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.
