Hunting (the word) game

While walking in the woods last week, I started wondering about the origin of the word game, as in animals that are hunted. Here's what I dug up in the OED: The etymology:

Old English gamen, gomen = Old Frisian game, gome, Old Saxon, Old High German gaman (Middle High German gamen), meaning joy, glee. Old Norse gaman (Swedish gamman, Danish gammen), meaning game, sport, merriment.

The word has seventeen definitions in the OED, but the first, ninth, and tenth are relevant here, and they give an idea of how the meaning of hunted animals came about:

1. Amusement, delight, fun, mirth, sport.

This is the oldest use documented by the OED, with citations from around the year 1000 and from 1160.

9. Obsolete. Sport derived from the chase. dog of game: one used in hunting or sporting. to be in game: to be engaged in the chase.

The OED's oldest quotation for this use is from 1297.

10a. The object of the chase; the animal or animals hunted.

The OED's first quote of this use is from 1486.