Game theory can be traced back to Babylonian times, in the so called Marriage Contract Problem, although this was only recognised as such in 1985. Game Theory really began to emerge as a subject in the 1900s. Firstly by Ernst Zermelo in his paper Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels ("On the Application of Set Theory to the Theory of Chess") of 1913, in which the first 'theorem' of game theory was published. This theory (known as Zermelo's Theorem) states that in the game of chess either white can force a win, or black can force a win, or both sides can force at least a draw.
The field came into being properly with the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern.