What I've written below was addressed to an interlocutor who believes that the decision to translate Revelation 5:10 as "over" rather than "on" is not a Greek but an English issue. I would like to see what the members of Bible Translation think. I wrote the foregoing to my interlocutor:
There are scholars who prefer to render EPI in Revelation 5:9-10 as "on" even though the context has reference to the authority of men and women, whom God (through Christ) has bought or redeemed from the earth.
Robert L. Thomas does not offer a justifying explanation for why he chooses to translate EPI as "on" rather than "over," but he does render this portion of Revelation 5:10: "and they shall reign on the earth" (Revelation 1-7: An Exegetical Commentary, page 402).
David Aune (in his Word Biblical Commentary on Revelation) prefers the translation "on" for EPI in Revelation 5:10. See his text Revelation 1-5, page 362. But this scholar also does not offer an explanation for his rendition of the verse.
However, both Charles B. Williams (New Testament in the Language of the People) and William F. Beck (New Testament in the Language of Today) choose to translate EPI as "over." I reproduce those renderings below:
"and they will rule over the earth" (Williams)
"and they will rule as kings over the earth" (Beck)
Is this issue simply one of how English treats [Greek] prepositions that function within the context of descriptions about authority or rulership? That is not what I glean from reading Greek grammars. Moreover, BDAG Greek-English lexicon (the authoritative NT Greek lexicon) states that EPI can be a "marker of power, authority, control of or over someone or [something], over." The examples that are listed in the lexicon include Revelation 5:10; 17:18, 20:6; 2:26. It doesn't seem like this is a matter of how English treats [Greek] prepositions within the context of authority (cf. Matthew 24:45-47; Acts 12:20; Ephesians 4:6).
Louw-Nida Greek-English Lexicon (semantic domain 37.9) also notes that
EPI (in particular contexts) can function as "a marker of the object
over which someone exercises a control or authority." This resource
suggests the rendering "over, with responsibility for." The examples
given in this work are Acts 8:27; Luke 1:33. Again, I do not see how
it is just a matter of English idiom. However, it is possible that I
am not seeing matters clearly. Nevertheless, let us consider some other sources.
William Douglas Chamberlain writes: "A metaphorical use [of EPI] with
the idea 'over,' in the sense of ruling, grows quite naturally out of
'upon': hO WN EPI PANTWN (Rom 9:5), 'the one who is over (rules) all
See _An Exegetical Grammar of the Greek New Testament_, pages 122-123.
"EPI with any of its cases can express the object of one's control,
authority, or rule. Jesus gave his disciples authority over the power
of the enemy (Luke 10:19 acc.)" is what one finds in Richard A.
Young's _Intermediate New Testament Greek_ grammar, page 98.
LSJ states that EPI can be used with the causal sense "over, of
persons in authority, EP' hOU ETAXQHMEN Hdt. 5.109; hOI [EPI] TWN
PRAGMATWN the public officers, D 18.247." See the entry for EPI (A.III.1).
Finally, I will quote Max Zerwick: "The accusative and the genitive are found together and in a quite similar sense in Mt 25,21: 'because thou wast faithful over little (EPI OLIGA) I will set thee over much (EPI POLLWN).' This example, however, belongs to the metaphorical use, where e.g. of rule 'over' we find in the NT, alongside the classical genitive, the accusative as in BASILEUSEI EPI TON OIKON IAKWB Lk 1,33." See Zerwick's _Biblical Greek, page 42.