“5324. Two Dogs fight for a Bone, and a third runs away with it.”
Introductio ad prudentiam: Part II (1727), Gnomologia (1732)
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Thomas Fuller (writer) 420
British physician, preacher, and intellectual 1654–1734Related quotes

"Confession" in Complete Works of Jack London, Delphi Classics, 2013
Variant: Charity is the bone shared with the dog, when you are just as hungry as the dog.

“[…] is running like a scolded dog! (usually said when a heel wrestler flees from a fight)”
Commentary Quotes
Source: Ross, Jim, http://www.wwe.com/superstars/smackdown/jimross/bio/, J.R.'s WWE Profile, 2008-01-06, January 5, 2008, JRsBarBQ.com

“5344. Valour would fight, but Discretion would run away.”
Compare Poor Richard's Almanack (1747) : Courage would fight, but Discretion won't let him.
Introductio ad prudentiam: Part II (1727), Gnomologia (1732)

“Those that look for bones are dogs.”
Poster against relatives of people forcibly disappeared during the military dictatorship. Cartaz contra desaparecidos do Araguaia irrita deputados http://politica.estadao.com.br/noticias/geral,cartaz-contra-desaparecidos-do-araguaia-irrita-deputados,378349. Estadão (28 May 2009).

“It's not the size of the dog in the fight; it's the size of the fight in the dog.”
Anonymous American proverb; since 1998 this has often been attributed to Mark Twain on the internet, but no contemporary evidence of him ever using it has been located.
Variants:
It is not the size of the dog in the fight that counts, but the fight in the dog that matters.
"Stub Ends of Thoughts" by Arthur G. Lewis, a collection of sayings, in Book of the Royal Blue Vol. 14, No. 7 (April 1911), cited as the earliest known occurrence in The Dictionary of Modern Proverbs, edited by Charles Clay Doyle, Wolfgang Mieder, and Fred R. Shapiro, p. 232
It is not the size of the dog in the fight that counts, but the fight in the dog that wins.
Anonymous quote in the evening edition of the East Oregonian (20 April 1911)
What counts is not necessarily the size of the dog in the fight — it's the size of the fight in the dog.
Dwight D. Eisenhower, declaring his particular variant on the proverbial assertion in Remarks at Republican National Committee Breakfast (31 January 1958) http://www.presidency.ucsb.edu/ws/index.php?pid=11229
Misattributed

“Two key rules of Third World travel: 1. Never run out of whiskey. 2. Never run out of whiskey.”
All the Trouble in the World (1994)

“The worst dog gets the best bone.”
Mesiras Nefesh, c. 1910. Alle Verk, vii. 155.

“Give a dog a bone, leave a dog alone. Let a dog roam and he'll find his way home.”
"Ruff Ryders' Anthem" https://www.youtube.com/watch?v=rpHtEa2II_s (1998), It's Dark and Hell Is Hot
1990s

“Always had more dogs than bones.”
Square One
Lyrics, Highway Companion (2006)