A Game-Theoretic Analysis of Updating Sets of Probabilities

TL;DR

This paper analyzes how an agent should update a set of probabilities using game theory, especially under the minimax decision criterion, revealing when to condition or ignore information and exploring calibration issues.

Contribution

It introduces a game-theoretic framework for updating sets of probabilities, clarifies conditions for conditioning or ignoring information, and links updating rules to calibration.

Findings

Characterizes when conditioning is optimal under minimax.

Explains anomalies like time inconsistency through game differences.

Connects conditioning with calibration in set-based uncertainty.

Abstract

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 important 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.