The purpose of this paper is to analyze how strategically the bilaterally monopolistic firms, only-one-seller and only-one-buyer, behave in a situation in which each firm has uncertain information on its opponent firm's cost. Even though the two firms know that seeking integrated profit leads to the optimized profit for both firms, each firm has an incentive to opportunistically behave to increase its share of the integrated profit. These opportunistic behaviors of the firms are analyzed through a game theoretic approach especially finding Nash equilibrium mixed strategies for the strategic profiles such as true-report or not and monitoring or not. The comparative statics to the Nash equilibrium mixed strategies shows that as the profit share increases the probability of monitoring an opponent firm is decreased while the probability increases as the size of the overstated production cost increases. This study also shows that high penalty and low monitoring cost lead to high probability to tell the truth of the production cost.