The obvious answer is for EK to drop the pilots bidding system altogether, and assign a rational/rotating roster of flights to each crew member, with the proviso that individual crew members could trade flights on a rational basis, keeping in mind the flying hours must be keep more or less equal.
Of course, rational is the key modus operandi in all this, so it might well not work at EK.
I have personally worked at two airlines where the above referenced pilots rostering system was in use, and most all seemed to like it quite well.
Pilots could trade flights up to a maximum of two per month, and junior pilots in each crew position also received benefits from operating to a variety of destinations....good and bad.