There's no special sauce for this. It's simply the most common sense and intelligent way I / we can think of running competitions fairly.
The competition judging process:
I, or the person doing the shortlisting for a particular competition, read all the entries and compile a shortlist.
Meantime, judges are encouraged to read as many entries as they can, and highlight any particular favourites they encountered to the person shortlisting.
That list is whittled down to a Top 10 which is then sent to a panel of judges. Occasionally, the quality will be so high enough that a top 12 or 15 is sent to the judges.
The panel is made up of a team of around ten or so people, including moderators, long-term members, and me.
Points are awarded via a simple 1 through 10 placement for each story. These points are then totalled to determine the winner.
If at the end of this process, points totals indicate a tie, the story which placed first most frequently will win (and so on if still tied).
I believe this is the fairest way to do it. It takes several days to read through the stories, score them, write notes and draw up a shortlist. Doing it this way eliminates people winning through popularity, or spamming etc.
No-one who enters a story competition is then involved in the judging of that competition.
Story verifiers are free to enter any competition, as are past winners. For the avoidance of doubt, competition stories or results are not discussed in any mod forum before the official public announcement is made. Story verifiers are appointed because they are good quality writers, so it's not unusual to see some of them place or win.
Usually, the top three stories will receive an Editor's Pick award, the only time they're ever awarded without being voted on by the usual EP panel. However, stories which place further down the list in a comp, or even don't place at all, can be nominated by any story verifier and referred to the EP panel for consideration, outwith the competition brief. Although there may be some overlap in judges, this is a separate panel and stories will be considered in their own right, the same as any other nominated story would be.
Thanks.