Here the two protocols each have their advantages. The Protocol 7 algorithm can carry on even if (1-p)DTauthorities simply refuse to cooperate. Once this many are corrupted, however, the entire election is invalidated and must be restarted, since not a single vote can be tallied.
In our protocol, a group of (1-p)DT authorities can still prevent any votes from being tallied, but only if these are distributed evenly across districts, in such a way that no district has enough cooperative authorities. A smaller set of only (1-p)T authorities in a single district is enough to prevent tallying of that district, but the cost of restarting that district's election is comparatively small. Thus our protocol is both more and less vulnerable than that of Protocol 7. On the one hand, it takes fewer corrupt authorities to damage the election, but the amount of damage is potentially smaller.
This can be seen more starkly in the comparison of Protocol 6 and subset authority voting. In Protocol 6, a single corrupt authority prevents any votes from being tallied, but in subset authority voting, a single corrupt authority disqualifies only the votes that it sees.