Green’s Dictionary of Slang

bulge n.

[SE bulge, a protuberance]

(US) an advantage.

[US](con. c.1840) ‘Mark Twain’ Huckleberry Finn 338: So he said, now for the grand bulge!
[UK]E.W. Hornung Amateur Cracksman (1992) 103: We had the bulge before; he has it now.
[UK]E.W. Hornung Black Mask (1992) 220: You have the bulge at present.
[Aus]Sun. Times (Perth) 16 Feb. 1/1: The Hindoo bulge on the night-caretaker business was badly punctured last week [when] the charge of assault on the white watchman blew out with a bang.
[US]S. Lewis Babbitt (1974) 173: Providing we can keep the bulge on ’em in membership.
[US]E.E. Cummings Enormous Room (1928) 167: They used what bulge they had to such good advantage that one of them, during my stay, was pursued with a revolver by their sergeant, captured, locked up, and shipped off for court-martial on the charge of disobedience and threatening the life of a superior officer.
[US]N.Y. Times 12 Oct. V. 1/8: Yale enjoyed a 10–0 bulge that did not begin to tell the story of its domination of the contest [DA].
J. Reichler Associated Press 8 Sept. n.p.: Brooklyn [baseball team] retained its 5½-game bulge over the N.Y. Giants [W&F].

In phrases

have the bulge on (v.) (also get the bulge on)

(US) to have an advantage over, to be in a superior position over.

[US]Spirit of the Times (N.Y.) 18 Dec. 498/3: Kate got the bulge on her at the start [DA].
[US]Richmond (VA) Enquirer 30 Nov. 4/5: It is in this respect that the South has ‘the bulge’ on the North, and will always have it.
[US]‘Mark Twain’ Innocents at Home 332: Well, you’ve ruther got the bulge on me.
[US] S.A. Mackeever, Glimpses of Gotham and City Characters 50/1: Most cities have only one coroner, and there’s where New York has got the ‘bulge’ on them.
[Aus]Bulletin (Sydney) 2 Oct. 9/2: Ah m ! It’s good to be the clerk from Lassetter’s. Solomon in all his glory hadn’t the ‘bulge on him’.
[UK] ‘’Arriet on Labour’ in Punch 26 Aug. 88/2: Mothers can’t strike, or unionise, or make demonstrations. The bloke ’as got the bulge on them.
[Aus]Truth (Sydney) 21 Jan. 2/6: The reason a dog can look so knowing is that he can’t say anything to spoil the effect. This is where the poodle has the bulge on the politician.
[US]C.L. Cullen Tales of the Ex-Tanks 119: If there’s anything more joyous in life than having the bulge on a desk Sergeant in a police station [etc.].
[UK]R. Beach Pardners (1912) 122: I guess I’ve got the bulge on you this time.
[US]‘O. Henry’ ‘Memoirs of a Yellow Dog’ in Four Million (1915) 115: I consider that I have the bulge on him as far as you could chase a rabbit.
[US]‘Tod Sloan’ Tod Sloan by Himself 293: A well-balanced horse getting away last or nearly last has frequently the ‘bulge’ on the others from the very beginning.
[US]E.E. Cummings Enormous Room (1928) 167: They had the bulge on everybody.
[US]S. Lewis World So Wide 25: We got the bulge on Europe not only in banking and university work.
[UK]Wodehouse Jeeves in the Offing 69: I had the bulge on the ordinary searcher.