I would play lotto: if I win more than 10M$, I take the black box and leave. Otherwise I’d look in the black box: if it is full, I also take the small one. If not, I leave with just the empty black box. As this should be inconsistent, assuming a time traveling Omega, it would either make him not choose me for his experiment or let me win for sure (assuming time works in similar ways as in HPMOR).
If I get nothing, it would prove the Omega wrong (and tell me quite a bit about how the Omega (and time) works). If his prediction was correct though, I win 11.000.000$, which is way better than either ‘standard’ variant.
I would play lotto: if I win more than 10M$, I take the black box and leave. Otherwise I’d look in the black box: if it is full, I also take the small one. If not, I leave with just the empty black box. As this should be inconsistent, assuming a time traveling Omega, it would either make him not choose me for his experiment or let me win for sure (assuming time works in similar ways as in HPMOR). If I get nothing, it would prove the Omega wrong (and tell me quite a bit about how the Omega (and time) works). If his prediction was correct though, I win 11.000.000$, which is way better than either ‘standard’ variant.
While that sounds clever at first glance:
We’re not actually assuming a time-traveling Omega.
Even if we were, he would just not choose you for the game. You’d get $0, which is worse than causal decision theory.