This work studies sequential assignment match processes, in which random offers arrive sequentially according to a renewal process, and when an offer arrives it must be assigned to one of given waiting candidates or rejected. Each candidate as well as each offer is characterized by an attribute. If the offer is assigned to a candidate that it matches, a reward R is received; if it is assigned to a candidate that it does not match, a reward r ≤ R is received; and if it is rejected, there is no reward. There is an arbitrary discount function, which corresponds to the process terminating after a random lifetime. Using continuoustime dynamic programming, we show that if this lifetime is decreasing in failure rate and candidates have distinct attributes, then the policy that maximizes total expected discounted reward is of a very simple form that is easily determined from the optimal single-candidate policy. If the lifetime is increasing in failure rate, the optimal policy can be recursively determined: a solution algorithm is presented that involves scalar rather than functional equations. The model originated in the study of optimal donor-recipient assignment in live-organ transplants. Some other applications are mentioned as well.