The Asymptotically Optimal Value of the Recentred N-Agent Cost of the Ising Population Model

Fix parameters as in The Ising Population Data (i), adopt the Ising population data with those parameters, and adopt The Ising Population Model Instantiates the Data of the Fluctuation Theory, The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal, The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity and The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form, with the numbers $\varrho_{\mathcal{A}}$, $c_{J}$, $\tilde{d}$, $\Delta$, $\Gamma$, the scalar solutions $z$ and $p$, and the families $Z$, $\Xi$ of those lemmas. Assume in addition that objects as in the paragraph The approximate Kalman family of The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States are given for the present data — an approximate Kalman policy $h^{\mathrm{K},N}$ at each level $N\ge1$ formed with the Riccati family $Z$ and the zero matrix in the role of $\Pi_{0}$, an $N$-agent driving system and a projected solution at each level, with recentred costs $\mathcal{J}^{\mathrm{K}}_{N}$ — and that they satisfy hypothesis (DK) of that theorem. Write $\mathcal{C}$ for the set of sequences of recentred costs of admissible families of solutions for these data, admissibility being as defined in that theorem through (I$'$), (CB) and (D0). Then the following hold.

1. (The theorem applies, and its value is explicit.)(i) The Ising population data together with the equilibrium triple $(S,A,P)$, the family $Z$ and the constants

$$\underline{b}=q_{0},\qquad \tilde{K}=0,\qquad \varrho_{\mathcal{A}}=\min\{1-\underline{a},\ \bar{a}-1\},\qquad c_{J}=\min\Bigl\{\psi,\ 2\mu,\ \frac{1}{4\chi}\Bigr\}$$

instantiate the common data and satisfy every standing hypothesis of The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States, namely (A), (U), (JC), (H2), (LipC), the optimality $[A]\in\mathcal{M}^{*}_{S_{0}}$, (TG), the standing hypothesis on $S^{*}=S$, and (OC), (X$'$), (H5). The family $Z$ is moreover the only one satisfying (H2). The matrices $\tilde{D}_{t}$, $W_{t}$ and $\Xi_{t}$ of that theorem are those of The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity and The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form, its Kalman covariance is $\Pi_{t}=p_{t}vv^{\top}$, and its real number $V^{*}$ is

$$V^{*}=\int_{[0,T]}\Bigl(4\,z_{t}+32\,\chi\,z_{t}^{2}\,p_{t}\Bigr)\,dt ,$$

the Lebesgue integral over the compact interval $[0,T]$ of a continuous integrand. In particular $V^{*}$ does not depend on the convexity weight $\mu$, and

$$0\ \le\ V^{*}\ \le\ T\,\frac{\Delta-1}{\chi}\Bigl(1+\frac{2(\Delta-1)}{\Gamma+2}\Bigr),\qquad \Delta=\sqrt{1+\chi\psi},\qquad \Gamma=\sqrt{4+2\tilde{d}} .$$

2. (Lower bound for every admissible family.)(ii) Let a family of solutions for these data be admissible, with recentred costs $\mathcal{J}_{N}$. Then for every real $\varepsilon>0$ there is a natural number $N_{\varepsilon}$ such that $\mathcal{J}_{N}\ge V^{*}-\varepsilon$ for every $N\ge N_{\varepsilon}$.

3. (The Kalman family attains it.)(iii) The Kalman family is admissible, each $\mathcal{J}^{\mathrm{K}}_{N}$ is a well-defined real number, the sequence $(\mathcal{J}^{\mathrm{K}}_{N})_{N\ge1}$ lies in $\mathcal{C}$ — so $\mathcal{C}$ is nonempty — and

$$\lim_{N\to\infty}\mathcal{J}^{\mathrm{K}}_{N}=V^{*} .$$

4. (The value.)(iv) $V^{*}$ is an asymptotically optimal value of $\mathcal{C}$ and is the only real number with this property; and it is the minimal linear-quadratic-Gaussian cost of the fluctuation problem, in the sense established in The Fluctuation LQG Value Is the Asymptotically Optimal Value of the Recentred N-Agent Cost over Admissible Families of Observation-Driven Policies, and Is Attained by the Approximate Kalman Policies, under Deterministic Initial States: for every linear-Gaussian state-observation model matched to the fluctuation LQG data as required there, the linear-quadratic-Gaussian cost formed from it attains a minimum over the extended admissible controls, and that minimum equals $V^{*}$.