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It is known that the capacity of the intelligent reflecting surface (IRS) aided cellular network can be effectively improved by reflecting the incident signals from the transmitter in a low-cost passive reflecting way. In this paper, we study the adoption of an IRS for downlink multi-user communication from a multi-antenna base station (BS). Nevertheless, in the actual network operation, the IRS operator can be selfish or have its own objectives due to competing/limited resources as well as deployment/maintenance cost. Therefore, in this paper, we develop a Stackelbeg game model to analyze the interaction between the BS and the IRS operator. Specifically, different from the existing studies on IRS that merely focus on tuning the reflection coefficient of all the reflection elements, we consider the reflection resource (elements) management, which can be realized via trigger module selection under our proposed IRS architecture that all the reflection elements are partially controlled by independent switches of controller. A Stackelberg game-based alternating direction method of multipliers (ADMM) is proposed to jointly optimize the transmit beamforming at the BS and the passive beamforming of the triggered reflection modules. Numerical examples are presented to verify the proposed studies. It is shown that the proposed scheme is effective in the utilities of both the BS and IRS.