On the Master Equation for Linear Quadratic Graphon Mean Field Games
Bibliographic record
Abstract
In this work a linear quadratic instance of Graphon Mean-Field Games (GMFGs) is analysed. Such games involve an asymptotically infinite population of agents, distributed over a very large scale network which itself is asymptotically infinite. The linear dynamics of each agent together with its quadratic running and terminal cost functions depend upon non-uniform averages (i.e. local and global mean fields) of the states of all other agents in the network system. In the infinite limit of the population and the network, the agent's dynamics and costs are functions of the family of local mean fields distributed at the nodes of the infinite network. Moreover, the limiting infinite networks are modelled by graphons which are symmetric measurable functions defined on the unit square. Specifically, Linear Quadratic Graphon Mean Field Games model the idea of clustering for populations of agents at the nodes of the very large scale network. First, using a probabilistic approach, we characterize the solutions of the Linear Quadratic Graphon Mean Field Games with solutions to coupled Forward Backward Stochastic Differential Equations (FBSDEs) of McKean-Vlasov type. We next deduce the existence of the so-called Master field, which allows for the decoupling of these FBSDEs. Finally, we derive the infinite dimensional Partial Differential Equation (PDE), so-called Master Equation, for which the Master Field is a solution.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".