In a 5- to 6-quart pan, bring 2 1/2 quarts water to a boil.
Meanwhile, peel potatoes and cut into 1 1/2-inch chunks. Add potatoes to boiling water, cover, and simmer until tender when pierced, about 20 minutes; drain.
Mash potatoes, adding aioli and enough broth to make potatoes creamy.
Stir in crab. Add salt and pepper to taste. Use in bouillabaisse or serve.