Green’s Dictionary of Slang

(US Und.) a form of confidence trick; see cit. 1869.

1869		N.Y. Times 10 July 2: The Envelope Game – Another Swindle. A number of complaints were made before Marshal Tooker yesterday by countrymen who have been swindled by what is known as the ‘envelope game.’ [A sucker] stated that he was induced to enter the basement of No. 2 Park-row, and there purchased tickets to the amount of $20, for which he was to choose enveloped containing orders for prizes. Mr. Vincent found, however, that the envelopes were blank, and hence concluded that he had been swindled.
1885		World (N.Y.) 25 Apr. 11/1–2: ‘Are there any more schemes for plucking the jays?’ ‘Lots! There’s the jewelry case and envelope game. You know all about them.’.
1900		Nat. Police Gaz. (NY) 24 Feb. 3/1: The famous envelope game is one of the biggest and best winners, and it flourishes best in the fake museums.