The name of the strategy game where players take turns to control armies is "chess." Chess is a two-player board game that involves strategic thinking and planning. Each player controls an army of 16 pieces, including pawns, knights, bishops, rooks, a queen, and a king, each with its own unique movement rules. The objective is to checkmate the opponent's king, meaning the king is in a position to be captured without escape. Chess has a rich history and has been studied extensively in terms of tactics and strategies, often fostering critical thinking and problem-solving skills in its players.