Two simple variations of top trading cycles

Top Trading Cycles is widely regarded as the preferred method of assigning students to schools when the designer values efficiency over fairness. However, Top Trading Cycles has an undesirable feature when objects may be assigned to more than one agent as is the case in the school choice problem. If agent 's most preferred object has a capacity of , and has one of the highest priorities at , then Top Trading Cycles will always assign to . However, until has the highest priority at , Top Trading Cycles allows to trade her priority at other objects in order to receive . Such a trade is not necessary for 's assignment and may cause a distortion in the fairness of the assignment. We introduce two simple variations of Top Trading Cycles in order to mitigate this problem. The first, Clinch and Trade, reduces the number of unnecessary trades but is bossy and depends on the order in which cycles are processed. The second, First Clinch and Trade, is nonbossy and independent of the order in which cycles are processed but allows more unnecessary trades than is required to be strategyproof and efficient. Both rules are strategyproof.