A proof of calibration via Blackwell's approachability theorem

Over the past few years many proofs of the existence of calibration have be en discovered. Each of the following provides a different algorithm and pro of of convergence: D. Foster and R. Vohra (1991, Technical Report, Universi ty of Chicago), (1998, Biometrika 85, 379-390), S. Hart (1995, personal com munication), D. Fudenberg and D. Levine (1999, Games Econ. Behavior 29, 104 -130), and S. Hart and A. Mas-Colell (1997, Technical Report, Hebrew Univer sity). Does the literature really need one more? Probably not. But the algo rithm proposed here has two virtues. First, it only randomizes between two forecasts that are very close to each other (either p or p + epsilon). In o ther words, the randomization only hides the last digit of the forecast. Se cond, it follows directly from Blackwell's approachability theorem, which s hortens the proof substantially. Journal of Economic Literature Classificat ion Numbers: C70, C73, C53. (C) 1999 Academic Press.