Ok, so there are N candidates. Choose a positive integer X. The system proceeds in a sequence of steps:
Step 1: A single tentative winning candidate is chosen uniformly at random. The voters can choose to accept or reject this candidate as the winner. If they are rejected, move on to the next step. Otherwise, they are declared the winner.
Step K for 1<K<=X-1: Choose a single random tentative winning candidate that is different from the candidate chosen in the previous step. The voters can choose to accept or reject this candidate. If they are rejected, move on to the next step. Otherwise, they are declared the winner.
Step X: Randomly choose a candidate not equal to the candidate chosen in Step X-1. If all candidates have been presented, the randomly chosen candidate on step X is declared the winner. Otherwise, the voters may accept or reject the candidate. If the candidate is rejected, repeat from Step 2. Otherwise, that candidate is the winner.
The way the voters accept or reject is up for discussion. Basically, voters are given many chances to compromise, and if they fail, they are punished with a random candidate as the winner.
My motivation for this kind of silly concept is known as the Secretary Problem:
It may also be modified in that the candidate chosen randomly at step K may be required to be different from the preceding L candidates. More generally, the probability of choosing a candidate could vary stepwise depending on whether or when they were chosen last.