Not that I endorse anything but a two-step procedure can enforce a kind of repeatibility.

  * Step 1. A thousand votes are selected randomly. And counted and the counts are made public
  * Step 2. The winner vote is selected at random in public procedure.

Now step 1 can be repeated at a contestant request, but if the counts are roughly the same, step 2 doesn't happen. Actually if the counts are too different, a revote may be in order