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etymonline.com wrote:game (n.)
c. 1200, from Old Englishgamen"joy, fun; game, amusement," common Germanic (cognates: Old Frisiangame"joy, glee," Old Norsegaman"game, sport; pleasure, amusement," Old Saxongaman, Old High Germangaman"sport, merriment," Danishgamen, Swedishgamman"merriment"), said to be identical with Gothicgaman"participation, communion," from Proto-Germanic*ga- collective prefix +*mann"person," giving a sense of "people together."
The -enwas 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 thegameinfair game(see underfair(adj.)), alsogamey. Meaning "number of points required to win a game" is from 1830.Game planis 1941, from U.S. football;game showfirst attested 1961.