What role do the brothers' given names play in contrast to Booth's adopted name "3-Card"?

Quick answer:

One could say that Booth is the younger brother and Lincoln is the older brother. Lincoln gives him a job and helps him, but Booth resents it. When he gets his chance to show off, he fails and then shoots Lincoln in anger.

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Their names set up the conflict and eventual resolution between them. 3-Card is the name that Booth adopts because he wants to be an expert at three card monte. Ultimately, however, Lincoln is still better and this conflict between them leads to his murder.

The names Booth and Lincoln set...

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the brothers against each other from the beginning. Just as John Wilkes Booth shot President Lincoln, Booth will eventually shoot his brother. Also, Lincoln is more positive and skilled while Booth is constantly failing at what he attempts and responds to these failures with rage. For example, near the end, he shoots the woman he wants to marry when she refuses him.

When Booth adopts the name 3-Card, it shows he wants to be better. It's not his birth name; it's something for him to aspire to. Lincoln teaches him and allows him some successes, but Lincoln is still the ultimately better player. When they compete for real money and Lincoln wins, Booth can't handle the humiliation. Lincoln gives him back his money but remarks on how easy winning against Booth was—and this leads Booth to shoot him.

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Even though the brothers were named as a joke by their father, their names prove to be the central image in the play. The names are obviously connected to history and establish the roles of each brother. Just as John Wilkes Booth assassinated Abraham Lincoln, we know Booth will kill his brother Lincoln. The tension is then established in the play, and we intensely watch and wonder what causes Booth to kill his brother.

Booth, the underdog, wants to be a hustler of three-card monte, and he constantly tries to prove he will be a better hustler than Lincoln if Lincoln will just show him how. This is why Booth invents his name of "3-Card". He has no sense of identity, of who he is. By allowing no one to call him anything but "3-Card", he boosts his confidence to be better than his brother.
Assuming his new identity of "3-Card" is the only way that Booth is able to kill his brother in the end.

Lincoln, the topdog, is satisfied with his job at the arcade. He was a former master of three card monte but gave up his hustling when his stick man was killed. He understands that in three-card monte, as in life, there is no winning. Booth never learns this, being overly confident and eager to prove himself the equal of his brother. When he sees Lincoln has hustled him, Booth can't take it and kills his brother.

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