Mathematical Methods in Data Economics: An Analytical Proof of Coincidence for Symmetric and Asymmetric AI Agents

This research examines the mathematical functional dependencies and specific marginal cost-price ratios of digital goods that govern the transition of an economic system from a hierarchical to a symmetric equilibrium. The study aims to derive the mathematical conditions for the convergence of the hierarchical Stackelberg model and the symmetric Cournot model within a data-driven economy, formalizing the cost and pricing parameters that induce leader-follower equivalence. The paper provides a theoretical framework and an analytical proof of market equilibrium transformation for digital products. It identifies a specific ratio of marginal costs to market price that neutralizes the first-mover advantage, thereby ensuring the convergence of leader and follower strategies. The mathematical equivalence of symmetric and asymmetric agents in Cournot and Stackelberg frameworks is validated under defined system constraints. These results establish an interdisciplinary basis for optimizing multi-agent systems (MAS) and verifying institutional shifts through computational mathematics, information theory, and software-based modeling. The practical scope of the findings includes the architectural design of high-performance MAS, cloud computing, and digital platform optimization. Furthermore, the analytical proof of hierarchical transformation offers a tool for antitrust regulation and technological sovereignty strategies, as well as for evaluating the long-term returns on market leadership. The study concludes that in an agent-based economy, traditional concepts such as information asymmetry, transaction costs, and market power are isomorphic to architectural parameters: API throughput, algorithmic response latency, and data interoperability protocols. Formalizing these isomorphisms enables a comprehensive computational model of the agent economy where institutional dynamics are verified via computational mathematics and computer linguistics.