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    Area of Science:

    • Game Theory
    • Mathematical Economics
    • Computational Economics

    Background:

    • Traditional mean-field game theory assumes a very large number of players, approximating systems with a continuum of decision-makers.
    • This continuum approximation is often impractical and meaningless for real-world networks with fewer entities.
    • Existing models are limited in their applicability to smaller-scale or sparsely populated systems.

    Purpose of the Study:

    • To develop a generalized mean-field framework suitable for systems of any size, from large to small.
    • To overcome the limitations of traditional mean-field approaches that rely on a continuum of players.
    • To provide a more versatile tool for analyzing strategic interactions in diverse network structures.

    Main Methods:

    • Development of a novel mean-field framework accommodating a finite number of entities.
    • Application of the framework to analyze dynamic auction models with asymmetric valuations.
    • Extension of the framework to incorporate scenarios with spiteful bidders.

    Main Results:

    • The proposed mean-field framework demonstrates applicability to systems with a small number of entities, unlike traditional models.
    • The framework successfully models complex scenarios such as dynamic auctions with heterogeneous participants.
    • Illustrative examples confirm the framework's utility in analyzing non-standard game-theoretic situations.

    Conclusions:

    • The generalized mean-field framework offers a significant advancement over traditional models by including small-scale systems.
    • This approach enhances the practical applicability of mean-field game theory in economics and other fields.
    • The framework provides a robust foundation for future research on strategic interactions in networks of varying sizes.