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From Nash Equilibrium to Social Optimum and Back: A Mean Field Perspective
René Carmona1, Gökçe Dayanıklı2, François Delarue3
1ORFE, Bendheim Center for Finance, Princeton University, Princeton, NJ 08544 USA.
Applied Mathematics and Optimization
|May 18, 2026
Summary
This study explores transitions between non-cooperative mean field games (MFGs) and cooperative mean field control (MFC) scenarios. It introduces new game classes and an algorithm for players to learn equilibria between these regimes.
Area of Science:
- Game Theory
- Mathematical Economics
- Computational Economics
Background:
- Mean field games (MFGs) and mean field control (MFC) model large populations in non-cooperative and cooperative settings, respectively.
- MFGs and MFCs approximate finite player problems with applications in economics, biology, and machine learning.
Purpose of the Study:
- Investigate transitions between non-cooperative (Nash equilibrium) and cooperative (social optimum) regimes in strategic player populations.
- Introduce and analyze novel game classes that bridge MFG and MFC frameworks.
Main Methods:
- Theoretical analysis of two new game classes: λ-interpolated MFGs and p-partial MFGs.
- Development of an algorithm for myopic players to learn p-partial mean field equilibria.
Main Results:
- Formalized the connections between non-cooperative and cooperative scenarios within mean field frameworks.
- Introduced λ-interpolated MFGs and p-partial MFGs to model intermediate behaviors between Nash equilibrium and social optimum.
Conclusions:
- The study provides a theoretical foundation for understanding regime shifts in strategic interactions.
- An algorithm is presented for learning equilibria in scenarios where populations partially deviate from cooperative behavior.
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