This paper investigates the non-parametric identification of matching games and applies this framework to the analysis of matching mechanism design in government recruitment. The empirical setting is the National Civil Service Exam (NCSE), a primary method for recruiting entry-level government officials in China. The NCSE is a mechanism where each candidate applies to one position and subsequently takes a meritocratic exam determining admission. An econometric challenge arises due to the absence of individual-level data in the NCSE dataset I collect. To overcome this challenge, I build a many-to-one Non-transferable Utility job matching model with two-sided heterogeneity for the labor market. I demonstrate its non-parametric identification using only position-level data, given instrument availability. Applying this framework to the NCSE, I assume that candidates differ in terms of their ability and civic-mindedness, with the latter being undetectable through the exam. After estimating the empirical model, I explore the sorting pattern induced by the NCSE. In the counterfactual analysis, I introduce a strategy-proof mechanism as an alternative. In this mechanism, following the meritocratic exam, each candidate chooses one position with higher-ranked candidates choosing first. Simulations show that candidates with similar types are more likely to be matched with similar jobs under the counterfactual mechanism compared to the NCSE. This is because the exam uncertainty inherent in the NCSE diminishes sorting. Finally, I explore the potential impact of different matching mechanisms on government performance by highlighting the pros and cons of adopting a “first apply, then exam” system (NCSE) or counterfactual mechanism in the context of multi-dimensional sorting, emphasizing the crucial influence of civic-mindedness.