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122 lines
3.2 KiB
Markdown
122 lines
3.2 KiB
Markdown
# Common-Noise Mean-Field Game Riccati Gate
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## Purpose
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Integrate **Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach** into the FAMM/Semantic Mass shortcut stack.
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The paper is useful because it gives a mathematically mature version of a move the project keeps making:
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```text
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many coupled local actors
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→ shared global field / common noise
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→ population-limit law
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→ reduced Riccati control kernel
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→ decentralized strategy
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→ bounded residual receipt
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```
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## Source
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- Wenyu Cong, Jingtao Shi, Bingchang Wang.
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- `Linear-Quadratic Mean Field Games with Common Noise: A Direct Approach`.
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- arXiv:2508.07271.
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## Why it matters
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The paper studies a linear-quadratic mean-field game with common noise where drift and diffusion terms are coupled with state, control, and mean-field state terms. It starts from a finite `N`-player game, derives FBSDEs by variational analysis, then takes the limit as `N → ∞` using law-of-large-numbers reasoning. In the limiting system, existence/uniqueness of BSDEs makes some variables identically zero, reducing the analysis enough to construct decentralized strategies with two Riccati equations. The paper also proves the constructed decentralized strategies form an `epsilon`-Nash equilibrium.
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## FAMM interpretation
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Common noise is a closure warning:
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```text
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if all agents share a shock,
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the shock is part of the system boundary.
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```
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The route object is not one local agent. It is the agent plus the population field plus the common-noise channel.
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## FAMM object
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```math
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\mathfrak C_{\mathrm{MFG}}
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=
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A_{16}(u_{\mathrm{mfg}})
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\otimes
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[
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\Sigma_{\mathrm{agent}}
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+
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\Sigma_{\mathrm{mean}}
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+
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\Sigma_{\mathrm{common}}
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+
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\Sigma_{\mathrm{FBSDE}}
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+
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\Sigma_{\mathrm{Riccati}}
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+
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\Sigma_{\epsilon\mathrm{Nash}}
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+
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\epsilon_{\mathrm{solv}}
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]
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```
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## Semantic Mass lanes
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```math
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\mu_{\mathrm{MFG}}[k]
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=
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w_m\|m_k\|
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+
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w_0\|W^0_k\|
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+
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w_u\|u_k\|
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+
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w_q J_k
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+
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w_r\|R_k\|
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+
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w_e\epsilon_{\mathrm{Nash},k}
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+
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w_s S_k
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```
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Where:
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- `m_k` = population / mean-field state.
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- `W0_k` = common-noise shock lane.
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- `u_k` = decentralized control intensity.
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- `J_k` = cost/value lane.
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- `R_k` = FBSDE/Riccati residual.
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- `epsilon_Nash,k` = bounded equilibrium error.
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- `S_k` = solvability/stability status.
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## 16D anchor addition
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```text
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COMMON_NOISE_MFG_RICCATI_GATE
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```
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Recommended axis placement:
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```text
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2 semantic mass population-state pressure
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3 Z pole mean-field trajectory recurrence
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4 curvature Riccati/value curvature
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5 delta-memory population state as compact history
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6 closure common-noise boundary inclusion
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7 residual seal epsilon-Nash residual
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11 scar failed coupling / unsolved FBSDE route
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13 invariant equilibrium consistency
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14 receipt Riccati/epsilon-Nash receipt
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```
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## Shortcut doctrine
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```text
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do not enumerate every strategic interaction
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when the population-limit field plus Riccati kernel carries the lawful structure.
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```
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## No-drift boundary
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This is a mathematical control/routing witness. It does not prove project-level claims by itself. It gives a rigorous shortcut pattern: common-noise closure, population-limit collapse, Riccati kernel, decentralized strategy, and bounded equilibrium residual.
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