Efficient and fair trading mechanisms for resource exchange in market design

We develop a method using parameterized linear equations to define trading mechanisms in market design models. Our method adeptly addresses challenges arising from factors such as complex endowments or coarse priorities, while offering flexibility to incorporate fairness concerns through the selection of equation parameters. Applying this method to models including fractional endowment exchange, priority-based allocation, and house allocation with existing tenants, we obtain new mechanisms that are both efficient and fair in these models.