Explanation in EASA FCL about how to fill out the EASA standard logbook:
When an aircraft carries two or more pilots as members of the operating crew, one of them shall, before the flight commences, be designated by the operator as the aircraft PIC, according to operational requirements, who may delegate the conduct of the flight to another suitably qualified pilot.
Which quite clearly implies that the operator (airline) designates the PIC before the flight commences, and the PIC can delegate the actual flying to a suitable qualified pilot. That means that during a flight the PIC will remain PIC, even if he is sleeping in the bunk and two other pilots are flying the aircraft. (see FCL.050).
The definition of PIC in FCL.010 it says quite clearly as well that the PIC is designated:
"Pilot-in-command" (PIC) means the pilot designated as being in command and charged with the safe conduct of the flight.
So in our business, we as pilots do not decide who is PIC, except if we are a management pilot for the operator. Crewing in the name of the operator usually does.