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.

💡 Research Summary

The paper investigates how an agent should revise her uncertainty when that uncertainty is represented not by a single probability distribution but by a set P of possible distributions, and the agent makes decisions according to the minimax (max‑min) criterion. The authors embed this decision problem in a two‑player zero‑sum game: the agent chooses an action after observing a realization x of a random variable X, while a “bookie” selects a concrete distribution Pr from the set P. Two natural versions of the game are distinguished. In the first version the bookie must commit to a distribution before the agent sees x (and therefore before the bookie knows the realized value). In the second version the bookie observes x and then picks a distribution, i.e., the bookie has full information about the signal when making his choice.

The paper shows that these two games generate fundamentally different optimal strategies for the minimax‑using agent, thereby explaining previously reported anomalies such as time‑inconsistency. In the “no‑signal‑knowledge” game the agent’s optimal minimax rule coincides with the rule she would use before any observation; the observed value x is effectively ignored because the worst‑case distribution the bookie can select does not depend on x. In the “signal‑knowledge” game the agent must first condition the set P on the observed value, forming the conditional set P|x = {Pr∈P : Pr(X=x)>0} and then apply the minimax criterion to this reduced set. Formally the optimal action a* solves

\