Since the game is finite (both players’ strategy sets are the finite setV⊆ H), von Neumann’s minimax theorem applies (von Neumann, 1928; Cesa-Bianchi and Lugosi, 2006)

In other words, for every adversary strategyqthere exists a learner strategyp(such asp=q) satisfyingPayoff(p, q)≤ 1 2, hence themaximinvalue of the game is at most 1 2 · 1928