The American-English phrase one may, or might, as well have the game as the name, and its variants, mean that, if one is falsely reputed to act in a specific manner, then one may as well act in that manner.

These are the earliest occurrences of the phrase that I have found, in chronological order:

1-: From The Scribbler, a Series of Weekly Essays, on Literary, Critical, Satirical, Moral, and Local Subjects; Interspersed with Pieces of Poetry (Montreal: Samuel Hull Wilcocke) of Thursday 23^rd January 1823:

For the Scribbler.
Card of Acknowledgment.

The two gentlemen from Upper Canada beg leave to present their thanks to the managers of the late rout at Fort Stark, as well for the civility and attention paid to them on their introduction, as for the polite manner in which they were taken leave of.
N. B. It is a rule with the Creamers, that one may as well have the game as the name, but it failed them, however, in this instance. They came Old Soldier over us, ran us hard, and rather crowded us. Yet who knows but an opportunity may erelong offer at Scant’s for sweet retaliation.

2-: From The Buffalo Convention and the Tariff, published in The Daily Union (Washington, D.C.) of Thursday 14^th September 1848:
—Context: During the presidential campaign of 1848, after the Whig Party and the Democratic Party nominated presidential candidates who were unwilling to rule out the extension of slavery into the western territories of the United States, anti-slavery Democrats and Whigs joined with members of the abolitionist Liberty Party to form the Free-Soil Party; Martin Van Buren (1782-1862), 8^th President of the USA from 1837 to 1841, was eventually nominated for the Free-Soil presidential candidacy:

Martin Van Buren, on the slavery question, has certainly changed his former expressed opinions; and on the question of a tariff for protection, no one will dispute us when we say he has changed. To be sure, he refers his friends to his Indiana letter for his views at that time, but the views there expressed do not accord with the resolution passed at Buffalo, and endorsed by Mr. Van Buren. The following is the resolution:
“Received, That the obligations of honor and patriotism require the earliest practicable payment of the national debt, and we are therefore in favor of such a tariff of duties as will raise revenue adequate to defray the necessary expenses of the federal government, and to pay annual instalments of our debt and the interest thereon.”
There is a tale to this, which we will explain: Joseph L. White ¹, a high protective tariff whig, but who was opposed to Gen. Taylor, and determined to have revenge on the whig party for not nominating Henry Clay, attended the Buffalo convention—as a spectator, provided he could not rule, but as a delegate, provided he could. He was invited to take part; which he declined at first. As it was well known that he was opposing Gen. Taylor, B. F. Butler ² was extremely anxious to secure him for Mr. Van Buren. He was invited to meet with the committee on resolutions, with the understanding that if they adopted his standard, he should join them. White said he could meet them on every question but the tariff. Butler gave him the privilege of drawing the resolution on that subject. Mr. White accepted, and drew the above resolution, with the addition of the words “give protection to our manufacturers,” after the word “government.” The word protection, Butler feared would grate harshly on the ears of the honest “barnburners,” ³ not that he cared anything about it, and Mr. White was induced to leave out the words above quoted—in other words, he was as completely “wooled” as were the abolitionists. Mr. White had no more idea of yielding his opinion on that subject, than the abolitionists had of yielding to the nomination of Mr. Van Buren, when they first arrived at Buffalo. The musical tones of Mr. Butler’s voice accomplished the thing before White was aware of it; and the abolitionists, innocent souls, having been drugged or thrown into a mesmeric stupor, and seduced into the embraces of the barnburners, and their virginity suspected, thought they might as well have the game as the name. Having been fairly seduced, and their character lost, all is now common, and “free-soil” is the “flash” word.

¹ In 1848, Joseph Livingston White (circa 1813-1861) opposed the Whig nomination of the slaveholder Zachary Taylor for President—Zachary Taylor (1784-1850) was the 12^th President of the USA from 1849 until his death.
² The lawyer Benjamin Franklin Butler (1795-1858) was a political ally of Martin Van Buren.
³ The term barnburner was a nickname of the radical section of the Democratic Party. According to Dictionary of Americanisms. A glossary of words and phrases, usually regarded as peculiar to the United States (New York: Bartlett and Welford, 1848), by the U.S. historian and linguist John Russell Bartlett (1805-1886), this term alludes “to the story of an old Dutchman who relieved himself of rats by burning his barns which they infested,—just like exterminating all Banks and Corporations to root out the abuses connected therewith.”

3-: From The Forged Will; or, Crime and Retribution (Philadelphia: T. B. Peterson, 1853), a novel by the U.S. author Emerson Bennett (1822-1905)—here, blame is used instead of name:

“As it is, she’ll think me a scoundrel, and so will Ellen. But no matter; I’ve been so considered all my life, and might as well have the game as the blame.”

4-: From Pen and Ink Photographs: The Medical and Surgical Profession of New York. Taken by an Amateur, published in The New York Atlas (New York City, N.Y.) of Sunday 24^th August 1856:

[The Doctor] strenuously asserts that he has never paid so much attention to diseases of the mind as he has the reputation for, and that he has become identified as a student of these eccentric complaints by chance, and not his own will. But with as little attention as he claims to have bestowed on them, we believe he could, at this time, write a better book on the medical jurisprudence of insanity than any yet produced—and we wonder that he does not. He may as well have the game as the name.

5-: From Slander, published in The Daily Evening Express (Lancaster, Pennsylvania) of Monday 27^th April 1857:

Many innocent persons have been led to live a life of sin, regardless of character or condition in life, by having false statements made about them. Thay [sic] say they “might as well have the game as the name.”

6-: From the New York Daily Times (New York City, N.Y.) of Friday 14^th August 1857:

A suit has been commenced by one Walter Williams, an ex-Policeman of the Second Ward, against the Editors of the Leader, for charging him with being an associate with panel thieves. […]
The following is a copy of the complaint:
Walter Williams vs. James H. Welsh, John Clancy and Henry Morford.—The complaint of Walter Williams against the above named defendants […] shows to this Court that the said defendants […] did on the first day of August, A. D. 1857, and on divers other days and times since that day, at the City and County of New-York, falsely, wickedly, and maliciously compose, write, and publish or cause to be written and published of and concerning the plaintiff the false, scandalous, malicious and defamatory libel in the New-York Leader, a weekly newspaper by that name published in the City of New-York, said libel purporting to be in the form of a police report in relation to a robbery having been committed at No. 39 Thomas-street, in said City, that is to say: “Walter Williams, Esq. […] It is possible that Walter Williams may have been unjustly accused, and it is equally probable that it would occur to his mind, after he had been deprived of his salary and virtually driven from the Police, that he might as well have the game as the name, and hence his decadence into Thomas-street.”

7-: From “Keep out of Politics”, published in the New-York Freeman’s Journal and Catholic Register (New York City, N.Y.) of Saturday 10^th October 1857:

We have for a considerable length of time been convinced that much injury has resulted to this country from the flippant assertion and repetition, that “politicians are a set of despicable rogues.” It has been reiterated until in part it has secured the fulfilment of the assertion. The honor belonging to this grave avocation of public life has been aspersed and destroyed; and honest men have avoided entering on the career, or having entered on it, and finding that in doing so they are set down as dishonest men, have given up, sometimes, their better inspirations, and concluded they may as well have the game as the name.