Before revision:  Dropping the graph-based definition of TTC , we use parameterized linear equations to describe how agents trade endowments or priorities at each step. Our mechanisms are ex-ante efficient. They satisfy various fairness axioms when parameters are properly chosen .
After revision:  Dropping the graph-based definition of TTC , we use parameterized linear equations to define trading algorithms. Our method has two advantages. First, our method avoids discussing the details of who trades with whom and how, which can be a difficult question in complex environments. Second, by controlling parameter values in our equations, our method is flexible and transparent to satisfy various fairness axioms when parameters are properly chosen .