Blackwell's Approachability Theorem

Blackwell's Approachability Theorem gives a condition under which a player in a repeated game with vector-valued payoffs can guarantee, through a suitable strategy, that the running average of those payoffs converges to a target set regardless of the opponent's play, generalizing the scalar case of guaranteeing a payoff at least some value. Named for David Blackwell, it underlies later work on regret minimization and calibrated forecasting, where the target set represents a bound on how far a forecaster's performance can fall short of a benchmark.