This paper investigates the non-parametric identification of matching games and applies this framework to the analysis of matching mechanism design in government recruitment. An econometric challenge in the empirical analysis of government recruitment is the frequent non-availability of individual-level data. To address this challenge, I build a many-to-one Non-transferable Utility job matching model with two-sided heterogeneity, where candidates apply for jobs and then face a meritocratic selection by employers. In equilibrium, a set of equilibrium hiring standards, informed by selection results, adjusts to clear the market. I demonstrate the non-parametric identification of the model using only position-level data, given the availability of instruments. I apply the model and identification result to analyze 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. 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. I compare sorting patterns under the two different mechanisms and explore the potential impact of different matching mechanisms on government performance in the context of multi-dimensional sorting.