They competed for it by giving speeches on why they were the bravest after Achilles to their Trojan prisoners, who, after considering both men's presentations, decided Odysseus was more deserving of the armour.
ðeɪ kəmpˈiːɾɪd fɔːɹ ɪt baɪ ɡˈɪvɪŋ spˈiːtʃᵻz ˌɔn wˌaɪ ðeɪ wɜː ðə bɹˈeɪvɪst ˈæftɚɹ ɐkˈɪliːz tə ðɛɹ tɹˈoʊdʒən pɹˈɪzənɚz, hˈuː, ˈæftɚ kənsˈɪdɚɹɪŋ bˈoʊθ mˈɛnz pɹˌɛzəntˈeɪʃənz, dᵻsˈaɪdᵻd ɑːdˈɪsɪəs wʌz mˈoːɹ dɪzˈɜːvɪŋ ʌvðɪ ˈɑːɹmɚ.