Etymology
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game (adj.1)
"lame," 1787, from north Midlands dialect, of unknown origin, perhaps a variant of gammy (tramps' slang) "bad," or from Old North French gambe "leg" (see gambol (n.)).
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video (adj.)

1935, as visual equivalent of audio, from Latin video "I see," first person singular present indicative of videre "to see" (see vision). As a noun, "that which is displayed on a (television) screen," 1937.

Engineers, however, remember the sad fate of television's first debut and are not willing to allow "video transmission" (as television is now called by moderns) to leave the laboratory until they are sure it will be accepted. [The Michigan Technic, November 1937]

video game is from 1973.

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video- 
word-forming element; see video (adj.).
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game (v.)
Middle English gamen "to sport, joke, jest," from Old English gamenian "to play, jest, joke;" see game (n.). The Middle English word is little recorded from c. 1400 and modern use for "to play at games" (1520s) probably is a new formation from the noun; and it might have been re-re-coined late 20c. in reference to computer games. Related: Gamed; gaming.
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game (n.)

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.

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game (adj.2)
"ready for action, unafraid, and up to the task;" probably literally "spirited as a game-cock," 1725, from game-cock "bird bred for fighting" (1670s), from game (n.) in the "sport, amusement" sense. Middle English adjectives gamesome, gamelich meant "joyful, playful, sportive."
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game-cock 
cock bred for fighting or from fighting stock, 1670s, from game (n.) in the sporting and amusement sense + cock (n.1). Figurative use by 1727.
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board-game (n.)
also boardgame, 1867, from board (n.1) + game (n.). Compare German Brettspiel.
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gaming (n.)
c. 1500, "gambling," verbal noun from game (v.). From 1980s in reference to video and computer games. Gaming-house is from 1620s; gaming-table from 1590s.
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cosplay (n.)

"practice or hobby of dressing as a character from a movie, book, or video game, especially one from Japanese manga and anime," 1993, according to Merriam-Webster, from costume (n.) + play (n.), based on a Japanese word formed from the same English elements and alleged to date from 1983. Also used as a verb.

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