Intelligent reflecting surface (IRS) has emerged as a promising and revolutionizing technology for future wireless networks. Most existing IRS studies focus on simple cooperative systems which usually have a single frequency band. In realistic non-cooperative multi-band networks, however, the existing IRS designs may be not applicable or have severe performance degradation. Thus, in the complex network environment, it is more rational to consider IRSs as public resources to be dynamically allocated to appropriate users. In this paper, we first introduce the auction theory to tackle the resource management problem for a multi-IRS-assisted non-cooperative network. An efficient auction algorithm framework is introduced to sub-optimally solve this non-convex problem. Simulation result illustrates that the significant performance improvement can be achieved by applying the auction algorithm in the complex multi-IRS-assisted non-cooperative network.