## 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 ${\mathcal A}$.