In the context of low inertia conditions in modern power systems, a fair allocation of inertia support responsibilities on the generation side can ensure equitable compensation for participants providing inertia related ancillary services. This not only incentivizes technological innovation but also reduces the overall cost of such services. Therefore, an intra-day economic dispatch model incorporating inertia compensation is proposed, and the allocation of inertia support responsibilities is explored using cooperative game theory. Specifically, the Aumann-Shapley method is applied to perform the allocation, with its axiomatic standards discussed in detail. Finally, the feasibility, axiomatic consistency, effectiveness, and adaptability of the proposed allocation method are analyzed using case studies based on the IEEE 30-bus system with different configurations. The results indicate that the Aumann-Shapley method is both rational and widely applicable. It satisfies allocation standards such as efficiency and fairness, effectively incentivizes generators to enhance their inertia support capabilities, and offers advantages over comparative methods in reducing production costs.