Depends. A convoluted counterexample:

Alice has to buy food for a party while Bob is sending invitations. Alice initially expects 10 guests to arrive and plans to buy corresponding amount of food. For simplicity let's suppose that having 1-person worth of food more than needed bears the same cost as having the same amount less than needed, and Alice's prior probability distribution is symmetric around 10, so she intends to buy 10 units of food.

But before going to the shop, she asks Bob how many people are coming. Bob is reluctant to answer, so Alice offers an arbitrary threshold 40. Actually only 15 guests are going to attend, but Bob doesn't know this. His prior expectations were more or less the same as Alice's, but by power of anchoring, he gives the answer "more than 40" with p=0.3. This answer causes Alice to buy 45 (the lowest round number greater than 40) units of food. If Bob answers "not more than 40", Alice will stick with her previous guess. Which means the cost is 0.3 * 30 + 0.7 * 5 = 12.5.

Without asking, Alice would buy 10 units of food and the cost of error would be only 5.