To supplement Qimmik a little.
You’ve got the essence of it, all right. This episode is not actually represented in the Iliad, however, since it comes only after Achilles’ death, which it still in the future (amazingly enough) by the time the Homeric poem ends. But the contest over Achilles’ arms (not just his shield) and its sequel did feature in early epic (the “Little Iliad,” now lost), as well as in Greek tragedy, which provided prototypes for the contest itself, and of course both speeches comprehensively deploy episodes from the Iliad—as well as from other Trojan saga. They give Ovid an opportunity to juggle any number of Iliadic and other mythological data, the reader’s knowledge of which is taken for granted (or rather Ovid challenges his readers to match his erudition).
It’s a given of the tradition that Odysseus/Ulixes wins, thanks more to his skill with words than to the merits of his case. Traditionally Ajax has the stronger claim, and is cheated out of the prize that was his due. Inasmuch as Ulixes could be considered Ovid’s alter ego in his rhetorical ingenuity and artistry, there is some piquancy in his presentation.
It makes sense that Ajax speaks first, not just because he’s the designated loser but because then Ulixes can employ his sophistries in countering the points that Ajax makes. Ulixes must necessarily trump Ajax.
Any Ovidian speech is going to be a pretty good one. Ajax’s textbook rhetorical competence is amusingly paradoxical given his traditional character (which he himself insists on) as a man of deeds not words—he’s supposed to be bluff, not rhetorically skilled. The genealogical relationship between himself and Achilles, for instance, or the over-elaborated idea that the weight of Achilles’ arms would only slow Ulixes down as he runs away—these are pure Ovid, quite out of character for Ajax. Ovid turns everyone into himself, even Ajax. But in the (predetermined) event, Ajax’s points all just play into Ulixes’ hands, and his relatively brief speech is completely swamped by Ulixes’ performance.
The introduction of Ulixes draws on what had become a famous description in the Iliad of him as a speaker—he looked unimpressive until he spoke, but when he spoke, his eloquence was irresistible, his words fell like snow-flakes in winter.
Interesting that Ovid writes Athena/Minerva and Ajax’s madness out of the story. He wants a stark and crisp human finale.
You’ll note when you get to it that Ovid contrives to conclude the episide with a metamorphosis—of Hyacinthus, who has no mythological connexion with any of this.
Enjoy!