Maybe I'm confused, but can't it just... pick one however it wants? As long as the final tally comes to "correct in 99.9% of cases" the problem is satisfied and it doesn't matter how it decided, does it? Why does it matter if there are muliple correct choices?

Edit: Nevermind, I see what you mean. If your algorithm depends on the Lotto, and Omega depends on your algorithm, then Omega depends on the Lotto. You can actually influence how often this "same number" scenario happens via your algorithm.