Loot priority. if he/she greed as soon as the MvP died and he got the loot it means its his loot even if you attack first and got MvP
from what i have observe Loot priority is like "The first person to deal a lot of damage" like for example the MvP has 1,200,000 HP and its a Amon Ra which stays at one spot.
You happen to attack first and dealt 100k damage on the first 1-2sec Amon Ra spawn but from 2-3secs he/she dealt a wooping 300k but stopped attacking it anymore so you dealt the rest of the damage. Your total damage to Amon Ra is 900k and his/her is 300k you got the MvP popup but you dont get the loot priority.
I don't know if this is really true... but from the way i see it on my Amon Ra hunt this always happens to me. i always try to make the lower level competitor do the most damage but ill do the highest damage in a few second... Dark Claw Amon Ra then CI it for 300-400k then let the lower level GX do the rest to give the MvP to him/her.