Blackwell Approachability

Sequentially a player decides to play \{p_t\}_{t=1}^\infty and his adversary decides \{q_t\}_{t=1}^\infty. At time t, a decision (p_t,q_t) results in a vector payoff A(p_t,q_t)\in {\mathbb R}^k. Given a_t is the average vector payoff at time t, Blackwell’s Approachability Theorem is a necessary and sufficient condition so that, regardless of the adversary’s decisions, the player makes the sequence of vectors \{a_t\}_{t=1}^\infty approach a convex set {\mathcal A}.

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