We test whether anchoring affects people's elicited valuations for a bottle of wine in individual decision making and in markets. We anchor subjects by asking them if they are willing to sell a bottle of wine for a transparently uninformative random price. We elicit subjects' Willingness-To-Accept for the bottle before and after the market. Subjects participate in a double auction market either in a small or a large trading group. The variance in subjects' Willingness-To-Accept shrinks within trading groups. Our evidence supports the idea that markets have the potential to diminish anchoring effects. However, the market is not needed: our anchoring manipulation failed in a large sample. In a concise meta-analysis, we identify the circumstances under which anchoring effects of preferences can be expected.