Unlike the Prince of Morocco, who preceded him in the play, the Prince of Arragon has no introductory scene in which the audience has an opportunity to learn about his character and personality. Arragon simply appears at Portia's house in Belmont, takes the oath required of every suitor who chooses among the three caskets, and proceeds directly to making his choice among the three caskets of lead, silver, and gold.
One of the functions that the Prince of Arragon serves in the play is to provide the audience with the actual conditions of the oath, which have not been fully stated previously in the play.
ARRAGON. I am enjoin'd by oath to observe three things:
First, never to unfold to any one,
Which casket 'twas I chose; next, if I fail
Of the right casket, never in my life
To woo a maid in way of marriage; Lastly,
If I do fail in fortune of my choice,
Immediately to leave you, and be gone.
(act 2, scene 9, lines 9–15)
Arragon seems to have no reservations about making the oath, and he's ready to make his choice. Through the process of his choice, the choice he makes, and his reaction to his reward, Arragon is shown as a coldly practical, unsentimental, unromantic, and supremely self-confident man.
Arragon immediately rejects the lead casket, which has the inscription "Who chooseth me, must give and hazard all he hath." Arragon has no intention of giving or risking all he has for a lead casket or simply to gain a wife.
As a side note, at no time in the scene does Arragon mention that he has any interest in Portia as a person, that he loves her, or that he's in any way attracted to her. The only time he addresses Portia personally is when he says, "Sweet, adieu" on his way out of the scene. For Arragon, this casket business and the possible acquisition of Portia for his wife seems like nothing more than a business transaction.
Arragon just as readily rejects the gold casket and its inscription, "Who chooseth me, shall gain what many men desire," because he doesn't wish to be thought of as one of "many men" of the "foolish multitude" nor rank himself with common men, the "barbarous multitudes."
The inscription on the silver casket, "Who chooseth me, shall get as much as he deserves," appeals to Arragon's arrogance, which is to say that it appeals to his inflated sense of his own self-importance and self-worth. Arragon convinces himself that he absolutely deserves Portia, and he chooses the silver casket.
Arragon is rewarded with what he deserves, which is a portrait, not of Portia but of a "blinking idiot" and a poem that calls him a fool who came to the choice of caskets with the head of a fool but that goes away with two foolish heads—his own and the portrait of the "blinking idiot."
Arragon makes no mention of losing Portia as his wife, nor does he seem particularly upset about that. Arragon simply can't believe that he chose the wrong casket. "Did I deserve no more than a fool's head?" he says. "Is that my prize? are my deserts no better?"
Arragon promptly leaves the scene with barely a nod to Portia, shaking his head in disbelief as he goes out the door. Portia dismisses him as a moth singed by a candle and has nothing more to say about him as she turns her attention to Bassanio, Portia's own choice for a husband, who has just appeared outside her door, intent on taking his own turn to choose among the caskets.