When a voter compares two lottery-lotteries, they take expected utilities with respect to the inner Δ, but they sample with respect to the outer Δ, and support whichever sampled thing they prefer.
If we collapse and treat everything like the outer Δ, that just gives us the original maximal lotteries, which e.g. is bad because it chooses anarchy in the above example.
If we collapse and treat everything like the inner Δ, then existence will fail, because there can be non-transitive cycles of majority preferences.
When a voter compares two lottery-lotteries, they take expected utilities with respect to the inner Δ, but they sample with respect to the outer Δ, and support whichever sampled thing they prefer.
If we collapse and treat everything like the outer Δ, that just gives us the original maximal lotteries, which e.g. is bad because it chooses anarchy in the above example.
If we collapse and treat everything like the inner Δ, then existence will fail, because there can be non-transitive cycles of majority preferences.