A Game-Theoretic Analysis of Updating Sets of Probabilities
  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.
