A specialized computer matching system programmed to match orders for financial instruments of any type; where orders are matched at a price equal to the best available price or within the best available prices in the market; where priority of matching orders is ranked by price; where orders at the same price may be further ranked based on other attributes; where orders may optionally specify that the order sender is willing to pay a fee to (or receive a rebate by) an Order Matching Entity (i.e., a fee/rebate contingency (FRC)); where an order's optional FRC must be met in order to match the order; and where an order which cannot be matched solely because its FRC rate prevents it from matching with another order temporarily yields priority—other than price priority—to other orders ranked below to permit those orders to be matched.