  We propose a new method to define trading algorithms in market design
environments. Dropping the traditional idea of clearing cycles in generated
graphs, 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 criteria. We
apply our method to several models and obtain new trading algorithms that are
efficient and fair.
