We hypothesise and confirm a previously unnoticed pattern within pre-existing data on the endowment effect, collected via seven experiments employing the original design. Subjects with low valuations in binary choice relative to other subjects set a proportionally higher willingness to accept. Those with high valuations set a proportionally lower willingness to pay. The results challenge current theories, including models of reference dependent preferences. The findings imply that buyers and sellers consider not only their own preferences, but also their perceptions of potential deals. We propose a model of optimal exchange that rationalises this behaviour and accounts for the new findings.