Old English gamen "game, joy, fun, amusement," common Germanic (cf. Old Frisian game "joy, glee," Old Norse gaman, Old Saxon, Old High German gaman "sport, merriment," Danish gamen, Swedish gamman "merriment"), regarded as identical with Gothic gaman "participation, communion," from Proto-Germanic *ga- collective prefix + *mann "person," giving a sense of "people together."
Meaning "contest played according to rules" is first attested c.1300. Sense of "wild animals caught for sport" is late 13c.; hence fair game (1825), also gamey. Game plan is 1941, from U.S. football; game show first attested 1961.