This paper proposes a new method to allocate the transmission fixed costs among the network participants in a pool-based electricity market. The allocation process relies on the circuit laws, utilizes the modified impedance matrix and is performed in two individual steps for the generators and loads. To determine the partial branch power flows due to the participants, the equal sharing principle is used and validated by the Shapley and Aumann-Shapley values as two preferred game-theoretic solutions. The proposed approach is also applied to determine the generators’ contributions into the loads, and a new concept, named circuit-theory-based equivalent bilateral exchange (EBE), is introduced. Using the proposed method, fairly stable tariffs are provided for the participants. Cross-subsidies are reduced and a fair competition is made by the proposed method due to the counter-flows being alleviated compared with the well-known Z-bus method. Numerical results are reported and discussed to validate the proposed cost allocation method. Comparative analysis reveals that the method satisfies all conditions desired in a fair and efficient cost allocation method. Finally, the developed technique has been implemented successfully on the 2383-bus Polish power system to emphasize that the method is applicable to very large systems.