The story of Bagaios and Oroetes appears in Herodotus 3.120-128 (web address 1). Oroetes, the Persian satrap of Sardis↗, Phrygia↗, and Ionia↗ (see omicron 645) crucified Polycrates, the client tyrant of Samos↗, because Polycrates would not acknowledge his overlordship. Oroetes, however, failed to fully support Darius I during the revolts which broke out at the time of his accession (521 BCE). Darius requested loyal Persians to volunteer to carry out the execution of Oroetes. Bagaios drew the lucky lot and carried the imperial order of execution to Sardis↗.
1For this part of the entry Adler cites Lexicon Ambrosianum 39.↩
2Herodotus 3.128.1 (web address 2); again (more fully) at pi 116.