Demand response (DR) aggregator controlling and aggregating flexible resource of residential users to participate in DR market will contribute the performance of DR project. However, DR aggregator has to face the risk that users may break the contract signed with aggregator and refuse to be controlled by aggregator due to the uncertainty factors of electricity consumption. Therefore, in this paper, community operator (i.e., DR aggregator) is proposed to equip auxiliary equipment, such as energy storage and gas boiler, to compensate for power shortage caused by users' breach behavior. DR aggregated resource with different auxiliary equipment will have different characteristics, such as breach rate of DR resource. In the proposed DR framework, for selling the aggregated resource, community operator has to compete the market share with other operators in day-ahead DR market. In the competition, each operator will try its best to make the optimal bidding strategy by knowing as much information of its opponents as possible. But, some information of community operator (e.g., DR resource's characteristic) belongs to privacy information, which is unknown to other operators. Accordingly, this paper focuses on the application of incomplete information game-theoretic framework to model the competition among community operators in DR bidding market. To optimize bidding strategy for the high profit with incomplete information, a Bayesian game approach is formulated. And, an effective iterative algorithm is also presented to search the equilibrium for the proposed Bayesian game model. Finally, a case study is performed to show the effectiveness of the proposed framework and Bayesian game approach.