Motivated by applications in online marketplaces such as ride-hailing, we study how strategic servers impact the system performance. We consider a discrete-time process in which, heterogeneous types of customers and servers arrive. Each customer joins their type's queue, while servers might join a different type's queue depending on the prices posted by the system operator and an inconvenience cost. Then the system operator, constrained by a compatibility graph, decides the matching. The objective is to design an optimal control (pricing and matching scheme) to maximize the profit minus the expected waiting times. We develop a general framework that enables us to analyze a broad range of strategic behaviors. In particular, we encode servers' behavior in a properly defined \emph{cost function} that can be tailored to various settings. Using this general cost function, we introduce a novel probabilistic fluid problem. The probabilistic fluid model provides an upper bound on the achievable net profit. We then study the system under a large market regime in which the arrival rates are scaled by $η$ and present a probabilistic two-price policy and a max-weight matching policy which results in a net profit-loss of at most $O(η^{1/3})$. In addition, under a broad class of customer pricing policies, we show that any matching policy has net profit-loss of at least $Ω(η^{1/3})$. To show generality of our framework, we present multiple extensions to our model and analysis. We conclude the discussion by presenting numerical simulations comparing different cost models and analyzing performance of the proposed pricing and matching policies.