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The classic secretary problem concerns the problem of an employer facing a random sequence of candidates and making online hiring decisions to try to hire the best candidate. In this paper, we study a game-theoretic generalization of the secretary problem where a set of employers compete with each other to hire the best candidate. Different from previous secretary market models, our model assumes that the sequence of candidates arriving at each employer is uniformly random but independent from other sequences. We consider two versions of this secretary game where employers can have adaptive or non-adaptive strategies, and provide characterizations of the best response and Nash equilibrium of each game.