c. 1200, from Old English gamen "joy, fun; game, amusement," common Germanic (cognates: Old Frisian game "joy, glee," Old Norse gaman "game, sport; pleasure, amusement," Old Saxon gaman, Old High German gaman "sport, merriment," Danish gamen, Swedish gamman "merriment"), said to be identical with Gothic gaman "participation, communion," from Proto-Germanic *ga- collective prefix + *mann "person," giving a sense of "people together."
The -en was lost perhaps through being mistaken for a suffix. Meaning "contest for success or superiority played according to rules" is first attested c. 1200 (of athletic contests, chess, backgammon). Especially "the sport of hunting, fishing, hawking, or fowling" (c. 1300), thus "wild animals caught for sport" (c. 1300), which is the game in fair game (see under fair (adj.)), also gamey. Meaning "number of points required to win a game" is from 1830. Game plan is 1941, from U.S. football; game show first attested 1961.
adjective suffix, "full of or characterized by," from Old English -ig, from Proto-Germanic *-iga- (source also of Dutch, Danish, German -ig, Gothic -egs), from PIE -(i)ko-, adjectival suffix, cognate with elements in Greek -ikos, Latin -icus (see -ic). Originally added to nouns in Old English; used from 13c. with verbs, and by 15c. even with other adjectives (for example crispy).