Equity in strategic exchange

New fairness notions aligned with the merit principle are proposed for designing exchange rules. We show that for an obviously strategy-proof, efficient and individually rational rule, (i) an agent receives her favorite object when others unanimously perceive her object the best, if and only if preferences are single-peaked, and (ii) an upper bound on fairness attainable is that, if two agents' objects are considered the best by all agents partitioned evenly into two groups, it is guaranteed that one, not both, gets her favorite object. This on the one hand reveals the importance of single-peaked preferences in a private economy and provides a support of "Gul's Conjecture", and on the other hand also indicates an unambiguous trade-off between incentives and fairness in design of exchange rules.