Before revision:  We propose a class of Fractional Top Trading Cycle mechanisms to solve allocation problems in which agents have fractional endowments or are ranked by coarse priorities . Dropping the graph-based definition of Top Trading Cycle , we use parameterized linear equations to describe how agents trade endowments or priorities  .
After revision:  We propose a class of Fractional Top Trading Cycle mechanisms to solve allocation problems in which agents have fractional endowments or are ranked by coarse priorities . Dropping the graph-based definition of Top Trading Cycle , we use parameterized linear equations to describe how agents trade endowments or priorities at each step .