  We explore the possibility of designing matching mechanisms that can
accommodate non-standard choice behavior. We pin down the necessary and
sufficient conditions on participants' choice behavior for the existence of
stable and incentive compatible mechanisms. Our results imply that
well-functioning matching markets can be designed to adequately accommodate a
plethora of choice behaviors, including the standard behavior that is
consistent with preference maximization. To illustrate the significance of our
results in practice, we show that a simple modification in a commonly used
matching mechanism enables it to accommodate non-standard choice behavior.
