The aggregators are intermediary players at the distribution system level. They manage the financial transactions of resources such as micro-turbines (MTs), photo-voltaic (PV) production systems, flexible demands (FDs), and must-run demands (MDs) in the energy market. Based on the regulation of many transmission system operators (TSOs) in European countries, an aggregator is obliged to be assigned to a balancing group (BG) represented by a balancing group manager (BGM). The TSO measures the energy imbalances at the BG level. Meanwhile, the BGM is responsible for the management of financial transactions between the aggregators and the TSO. The strategic aggregators can establish a common BG instead of individual BGs. Then, they gain more payoffs since their energy imbalances are compensated more efficiently. However, the BGM should have the complete knowledge of the aggregators' confidential information, which includes the evaluation of aggregators concerning the cost/utility functions of associated resources. The aggregators have no incentive to disclose this valuable information. To solve this challenge, an iterative distributed algorithm based on non-cooperative game theory is proposed. In the numerical results, the performance of proposed common BG has been evaluated thanks to the comparison with a benchmark case where the aggregators establish their own individual BGs.