A Geometric Proof of Calibration

We provide yet another proof of the existence of calibrated forecasters; it has two merits. First, it is valid for an arbitrary finite number of outcomes. Second, it is short and simple and it follows from a direct application of Blackwell's approachability theorem to a carefully chosen vector-valued payoff function and convex target set. Our proof captures the essence of existing proofs based on approachability (e.g., the proof by Foster (1999) in the case of binary outcomes) and highlights the intrinsic connection between approachability and calibration.