Nash, Social and Centralized Solutions to Consensus Problems via Mean Field Control Theory
Bibliographic record
Abstract
The purpose of this paper is to synthesize initial mean consensus behavior of a set of agents from the fundamental optimization principles of i) stochastic dynamic games, and ii) optimal control. In the stochastic dynamic game model each agent seeks to minimize its individual quadratic discounted cost function involving the mean of the states of all other agents. In this formulation we derive the limiting infinite population mean field equation system and explicitly compute its unique solution. The resulting mean field (MF) control strategies drive each agent to track the overall population's initial state distribution mean, and by applying these decentralized strategies, any finite population system reaches mean consensus asymptotically as time goes to infinity. Furthermore, these control laws possess an εN-Nash equilibrium property where εNgoes to zero as the population size N goes to infinity. Finally, the analysis is extended to the case of random mean field couplings. In the social cooperative formulation the basic objective is to minimize a social cost as the sum of the individual cost functions containing mean field coupling. In this formulation we show that for any individual agent the decentralized mean field social (MF Social) control strategy is the same as the mean field Nash (MF Nash) equilibrium strategy. Hence MF-Nash ControlsUNashN=MF - Social ControlsUSocN. On the other hand, the solution to the centralized LQR optimal control formulation yields the Standard Consensus (SC) algorithm whenever the graph representing the corresponding topology of the network is Completely Connected (CC). Hence Cent. LQR ControlsUCentN=SC-CC ControlsUSCN. Moreover, a system with centralized control laws reaches consensus on the initial state distribution mean as time and population size N go to infinity. Hence, asymptotically in time MF-Nash ControlsUNashN=MF-Social ControlsUSocN= Cent. LQR ControlsUCent∞= SC-CC ControlsUSC∞. Finally, the analysis is extended to the long time average (LTA) cost functions case.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".