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Econ-28 · economics · Economics

Large-population selection of finite-state potential minimizers

LLM status Solved unverified

Statement

GT-084 | Mean-field games | P: published selection question; source-date status only

Definitions and mathematical statement

Let $E=\{1,\ldots,d\}$, $d\geq3$, $\Delta_d=\{m\in[0,1]^d:\sum_i m_i=1\}$, $T>0$, and $m_0\in\operatorname{int}\Delta_d$. Fix $C^3$ potentials $\mathcal F,\mathcal G$ on a neighborhood of $\Delta_d$, and set \[ f_i(m)=\partial_{m_i}\mathcal F(m),\qquad g_i(m)=\partial_{m_i}\mathcal G(m). \] Using extensions merely specifies component representatives: tangent differences of these gradients determine the rates on the simplex.

An admissible deterministic control is a measurable rate matrix $a(t)$ with $a_{ij}\geq0$ for $j\ne i$ and $a_{ii}=-\sum_{j\ne i}a_{ij}$, together with an absolutely continuous law $m:[0,T]\to\Delta_d$ satisfying \[ \dot m_i=\sum_{j\ne i}(m_j a_{ji}-m_i a_{ij}),\qquad m(0)=m_0, \qquad \int_0^T\sum_{i}\sum_{j\ne i}m_i a_{ij}^2dt<\infty. \] Its potential action is \[ \mathcal J(m,a)=\int_0^T\left[ \tfrac12\sum_i\sum_{j\ne i}m_i(t)a_{ij}(t)^2+\mathcal F(m(t))\right]dt +\mathcal G(m(T)). \] Let $\mathcal M_{T,m_0}$ be the set of law paths $m$ occurring in global minimizers of this action, with the same initial condition. This is a cooperative potential functional, not $\sum_i m_if_i$ in place of $\mathcal F$.

In the associated $N$-player game, player $\ell$ has state $X^\ell\in E$ and controls its jump rates from its current state $i$ to $j\ne i$. Let $m_t^{N,-\ell}=(N-1)^{-1}\sum_{r\ne\ell}\delta_{X_t^r}$. Its cost is \[ \mathbb E\!\left[\int_0^T\left( \tfrac12\sum_{j\ne X_t^\ell}|a_{X_t^\ell j}^{\ell}(t)|^2 +f_{X_t^\ell}(m_t^{N,-\ell})\right)dt +g_{X_T^\ell}(m_T^{N,-\ell})\right]. \] Admissible finite-player rates are bounded progressively measurable matrices with nonnegative off-diagonal entries and row sums zero. Individual transition noises are independent, and there is no common noise. Consider bounded symmetric Markov-perfect Nash feedbacks: at every time and state vector, no player can reduce its continuation cost by changing its own admissible rates while the other feedback functions stay fixed. Initial states are independent with common law $m_0$ and independent of future transition noises. Write $\mu_t^N=N^{-1}\sum_\ell\delta_{X_t^\ell}$.

Question. If along any sequence of these equilibria \[ \mathcal L(\mu^N)\Longrightarrow Q \quad\text{in }D([0,T],\Delta_d) \quad\text{and}\quad Q(C([0,T],\Delta_d))=1, \] must $Q(\mathcal M_{T,m_0})=1$? The topology on $D$ is Skorokhod $J_1$. A deterministic limit is the special case $Q=\delta_m$. The question concerns globally minimizing paths, rather than merely first-order equilibria.

Short English statement

Must every equilibrium selected by the large finite game minimize the corresponding potential control problem when there are at least three states?

Variants and scope boundaries

The deterministic MFG conditions are \[ -\dot u_i+\tfrac12\sum_{j\ne i}(u_i-u_j)_+^2=f_i(m),\quad \dot m_i=\sum_{j\ne i}[m_j(u_j-u_i)_+-m_i(u_i-u_j)_+],\quad u_i(T)=g_i(m(T)). \] Here $r_+=\max(r,0)$. These are necessary optimality/equilibrium conditions, not a global-minimum test. The source reports a positive two-state selection result; finite-player common noise, open-loop equilibria, and arbitrary approximate equilibria are separate variants.

Source relationship and status

Section 4.3 explicitly separates the open $d\geq3$ large-$N$ question from its two-state case. The path-law formulation allows randomized subsequential selection. No exhaustive later-literature audit is claimed.

References

[S1] Cardaliaguet and Delarue. Selected topics in mean field games, ICM2022, vol. 5; DOI: 10.4171/ICM2022/17.

Locators: (4.7)--(4.10), pp. 3683--3684, rate and equilibrium definitions; (4.18), p. 3689, potential condition; (4.19), Proposition 4.4 and "Selection of equilibria," p. 3690, action and explicit $|E|\geq3$ question.

https://ems.press/content/book-chapter-files/33265

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Econ-28-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-28

Research attempt and adversarial verification

9 October 2026 Original-target status: UNRESOLVED

C73-27: Large-population selection of potential minimizers

Input: C73-27.tex.  Tools: web browsing [YES]; Python computation [YES].

1) FORMAL RESTATEMENT

Fix $d\geq3$, $T>0$, interior $m_0\in\Delta_d$, and smooth potentials $\mathcal F,\mathcal G$ on a neighborhood of the simplex, with $f_i=\partial_i\mathcal F$, $g_i=\partial_i\mathcal G$. A deterministic population control consists of an absolutely continuous simplex path $m$ and measurable rates $a_{ij}\geq0$ ($i\ne j$), with \[ \dot m_i=\sum_{j\ne i}(m_ja_{ji}-m_ia_{ij}),\quad m(0)=m_0, \quad J(m,a)=\int_0^T\left[\tfrac12\sum_{i,j\ne i}m_i a_{ij}^2+ \mathcal F(m)\right]dt+\mathcal G(m(T)). \] Finite energy means finiteness of the displayed quadratic-rate integral. Let $\mathcal M_{T,m_0}$ be the paths occurring in global minimizers of $J$.

For each integer $N\geq2$, player $\ell$ controls the nonnegative jump rates of its current state, with diagonal rate equal to minus their sum. Its running and terminal costs are \[ \tfrac12\sum_{j\ne X_t^\ell}(a_{X_t^\ell j}^\ell(t))^2+ f_{X_t^\ell}(m_t^{N,-\ell}),\qquad g_{X_T^\ell}(m_T^{N,-\ell}),\quad m_t^{N,-\ell}=\frac1{N-1}\sum_{r\ne\ell}\delta_{X_t^r}. \] Individual transition noises are independent, with no common noise. Initial states are independent with law $m_0$. Equilibria are bounded symmetric Markov-perfect feedbacks: from every time and state vector, no unilateral bounded progressively measurable rate deviation can reduce that player's continuation cost. The bound on a feedback need not be uniform in $N$.

The literal question is whether, for every such datum and every sequence of such equilibria whose empirical laws $\mathcal L(\mu^N)$ converge in Skorokhod $J_1$, with continuous-path limit law $Q$, one has $Q(\mathcal M_{T,m_0})=1$, where $\mu^N=N^{-1}\sum_\ell\delta_{X^\ell}$. The minimal convention $N\geq2$ is necessary because $m^{N,-\ell}$ is undefined at $N=1$; it does not change the limit question. Exact Markov perfection, ex ante approximate Nash, first-order MFG conditions, and global potential optimality are distinct requirements.

2) QUICK LITERATURE/CONTEXT CHECK

Cardaliaguet and Delarue [cardaliaguet-delarue], Section 4.3 and the "Selection of equilibria" paragraph on printed p. 3691, distinguish the $d\geq3$ large-$N$ selection question from the two-state result and from vanishing-noise selection with corrected costs. The checked source does not justify replacing global minimization by the deterministic equilibrium equations. The present note supplies direct proofs of compactness, an affine-potential solved subclass, and an explicit failure for an ex ante approximate-equilibrium variant. The targeted check was performed on 9 October 2026, not asserted to be an exhaustive survey.

3) ATTACK PLAN

Type and tools. This is a many-player equilibrium limit with a variational selection requirement. Useful tools are: flux variables (compactness of the cooperative problem); convex duality of quadratic perspectives (lower semicontinuity); Dynkin's formula (single-agent verification); energy bounds from a zero-rate deviation (uniform estimates for exact equilibria); empirical martingales (large-$N$ compactness); Bregman remainders (separate convexity from stationarity); explicit concave terminal potentials (construct nonminimizing equilibria).

Proof tracks. Derive general compactness and an exact global verification identity; solve affine potentials completely. Disproof tracks. Try the zero-control nonminimizing MFG path in a finite game, then test it at every configuration; compare exact and approximate equilibrium quantifiers. The zero profile fails exact Markov perfection, but yields a rigorous counterexample to a weaker ex ante approximate-Nash selection statement.

4) WORK

Lemma (Existence and compactness of global minimizing paths): The deterministic potential problem has a minimizer. Its set $\mathcal M_{T,m_0}$ is compact in the uniform topology on $C([0,T],\Delta_d)$, hence is Borel.
Proof: Set $w_{ij}=m_i a_{ij}\geq0$. The energy becomes \[ K(m,w)=\frac12\int_0^T\sum_{i,j\ne i}\frac{w_{ij}^2}{m_i}\,dt, \] with $0/0=0$ and $w^2/0=+\infty$ when $w>0$. The path constraint is linear: $\dot m_i=\sum_{j\ne i}(w_{ji}-w_{ij})$. A constant path with zero flux is feasible. Boundedness of the potentials on the simplex implies that a minimizing sequence has uniformly bounded energy. Since $m_i\leq1$, its fluxes are bounded in $L^2$. The forward equation and Cauchy--Schwarz give a uniform $1/2$-Hölder modulus for the paths. Compactness of the simplex and the Arzelà--Ascoli theorem yield a uniformly convergent subsequence $m^n\to m$; weak compactness of bounded sets in the Hilbert space $L^2$ yields $w^n\rightharpoonup w$ along a further subsequence. Testing against time-interval indicators passes the integrated forward equation to the limit. Nonnegativity of the fluxes is preserved by weak convergence, since their integrals against every nonnegative $L^2$ test remain nonnegative.

For completeness, lower semicontinuity follows from \[ K(m,w)=\sup_{q\in L^\infty} \int_0^T\sum_{i,j\ne i}\left[q_{ij}w_{ij}-\tfrac12m_iq_{ij}^2\right]dt. \] The pointwise quadratic identity gives "$\geq$" for the supremum representation. For the reverse approximation use nonnegative truncated maximizers $q_{ij}=\min\{L,w_{ij}/m_i\}$, assigning $L$ where $m_i=0<w_{ij}$ and zero where both vanish. The integrands increase to the perspective as $L\to\infty$, so monotone convergence gives equality, including the infinite case. For each fixed bounded $q$, its expression is continuous under uniform $m$ and weak $L^2$ flux convergence. Taking their supremum proves the claimed lower semicontinuity.

The potential terms converge by uniform continuity. The limiting pair therefore minimizes. Finite energy forces $w_{ij}=0$ almost everywhere where $m_i=0$; define $a_{ij}=w_{ij}/m_i$ where $m_i>0$, and zero otherwise, to recover an admissible rate representation. The same compactness argument applied to any sequence of minimizers shows that every uniform limit extracted is a minimizing path. Thus $\mathcal M_{T,m_0}$ is compact. ∎

Theorem (General compactness of empirical laws of exact equilibria): For the finite-player games in the input, every sequence of bounded exact Nash feedback profiles (in particular every specified Markov-perfect sequence) has tight empirical path laws in $D([0,T],\Delta_d)$ with $J_1$ topology, and every subsequential limit is concentrated on continuous paths starting at $m_0$.
Proof: Let $f_{\min},f_{\max}$ be the minimum and maximum of all $f_i(m)$ on the compact simplex, and define $g_{\min},g_{\max}$ likewise. A player can deviate to never jumping. Regardless of how the other feedbacks react, that deviation costs at most $Tf_{\max}+g_{\max}$. Exact Nash therefore bounds the player's equilibrium cost by this number. Subtracting the lower bounds on its running and terminal interaction costs gives \[ \mathbb E\int_0^T\sum_{j\ne X_t^\ell} (a_{X_t^\ell j}^\ell(t))^2dt \leq C:=2[T(f_{\max}-f_{\min})+g_{\max}-g_{\min}], \tag{E} \] uniformly in $\ell,N$. Boundedness of each finite-game rate profile makes all compensators below well defined; (E) supplies the uniform estimates.

Write the empirical semimartingale decomposition as \[ \mu^N(t)=\mu^N(0)+\int_0^t B^N(s)ds+M^N(t). \] Let $A_N(t)=N^{-1}\sum_\ell\sum_{j\ne X_t^\ell}(a_{X_t^\ell j}^\ell(t))^2$. Each jump contributes opposite signs to two coordinates, so \[ \|B^N(t)\|_1\leq\frac2N\sum_{\ell,j\ne X_t^\ell}a_{X_t^\ell j}^\ell(t), \qquad \mathbb E\int_0^T\|B^N(t)\|_1^2dt\leq4(d-1)C. \] For the second assertion use Cauchy--Schwarz over the $N(d-1)$ current outgoing rates and then (E).

Independent individual jump noises have no simultaneous jumps almost surely. A jump changes the squared Euclidean empirical increment by $2/N^2$. Thus the sum of the coordinate martingale brackets obeys \[ \mathbb E\sum_i\langle M_i^N\rangle_T \leq\frac2{N^2}\sum_\ell\mathbb E\int_0^T \sum_{j\ne X_t^\ell}a_{X_t^\ell j}^\ell(t)dt \leq\frac{2\sqrt{T(d-1)C}}N. \] Doob's inequality on each coordinate gives \[ \mathbb E\sup_{t\leq T}\|M^N(t)\|_2^2 \leq\frac{8\sqrt{T(d-1)C}}N\longrightarrow0. \] Let $Y^N(t)=\mu^N(0)+\int_0^tB^N(s)ds$, a continuous path in $\mathbb R^d$. With arbitrarily high probability its squared-derivative integral is bounded by a deterministic constant, by Markov's inequality and the uniform energy bound. Such paths, with initial points in the compact simplex, form a set with compact closure in the uniform topology: Cauchy--Schwarz bounds increments by a constant times $|t-s|^{1/2}$ and bounds their suprema. Therefore the laws of $Y^N$ are tight on continuous paths. Since $\sup_t\|\mu^N-Y^N\|_2\to0$ in probability, the empirical laws are tight in $J_1$ and every limit is continuous. One can see the latter assertion directly by extracting a weakly convergent subsequence of the tight continuous approximations and using their vanishing uniform discrepancy; uniform distance dominates $J_1$ distance. Finally, the empirical initial law satisfies $\mathbb E\|\mu^N(0)-m_0\|_2^2\leq1/N$. Continuity of time-zero evaluation and the closedness of the simplex imply the stated initial point and path range for every limit. ∎

What compactness does not establish.
This theorem supplies the kind of subsequences postulated in the question without assuming a uniform bound on all feedback rates. It does not identify their limiting controls, prove that they solve a deterministic MFG pathwise, or prove global potential optimality. Those are separate steps.
Lemma (Exact potential verification identity): Let $(m^*,u)$ be a bounded classical deterministic MFG solution with \[ \dot u_i=H_i(u)-f_i(m^*),\quad u_i(T)=g_i(m^*(T)),\quad a^*_{ij}=(u_i-u_j)_+,\quad \dot m^*=\operatorname{div}(m^*a^*). \] For any admissible finite-energy competitor $(m,a)$ with the same initial law, define $D_{\mathcal F}(m,m^*)=\mathcal F(m)-\mathcal F(m^*)-f(m^*)\cdot(m-m^*)$, and define $D_{\mathcal G}$ analogously. Then \begin{align*} J(m,a)-J(m^*,a^*)={}&\int_0^T\sum_{i,j\ne i}m_i \left[\tfrac12(a_{ij}-(u_i-u_j)_+)^2+(u_j-u_i)_+a_{ij}\right]dt\\ &+\int_0^TD_{\mathcal F}(m,m^*)dt +D_{\mathcal G}(m(T),m^*(T)). \end{align*}
Proof: Finite energy implies integrable fluxes: $\int_0^Tm_i a_{ij}dt\leq \sqrt{T\int_0^Tm_i a_{ij}^2dt}$, since $m_i\leq1$. Integration by parts with $u\in C^1$ and the absolutely continuous population paths is therefore justified. The initial difference is zero, and hence \[ g(m^*(T))\cdot(m(T)-m^*(T)) =\int_0^T\left[(H(u)-f(m^*))\cdot(m-m^*) +u\cdot(\dot m-\dot m^*)\right]dt. \] For a flux $(m_i a_{ij})$, the last linear term is $u\cdot\dot m=\sum_{i,j\ne i}m_i a_{ij}(u_j-u_i)$. Expand the potential differences as their linear parts plus $D_{\mathcal F},D_{\mathcal G}$ and substitute this integration-by-parts identity. The terms linear in $f(m^*)$ cancel. Writing $z=u_i-u_j$, the remaining scalar identity is \[ \tfrac12a^2-za+\tfrac12z_+^2 =\tfrac12(a-z_+)^2+(-z)_+a. \] For $a=z_+$ the expression is zero. Summing over the competitor fluxes and subtracting the zero expression for $(m^*,a^*)$ proves the displayed identity exactly. ∎
Proposition (Convex potentials turn classical equilibrium into global optimality): If $\mathcal F$ and $\mathcal G$ are convex on the simplex, every bounded classical MFG solution as above is a global potential minimizer, and its population path is the unique minimizing path.
Proof: Convexity implies $D_{\mathcal F},D_{\mathcal G}\geq0$ by the first-order convexity inequality on each line segment in the simplex. Every other term in the identity is nonnegative because rates and masses are nonnegative. Thus $J(m,a)\geq J(m^*,a^*)$ for every admissible competitor. Equality forces $m_i(a_{ij}-a^*_{ij})^2=0$ almost everywhere, and hence $m_i a_{ij}=m_i a^*_{ij}$ almost everywhere. The competing population then solves the same bounded linear forward ODE with rates $a^*(t)$ and the same initial law. Uniqueness follows from Gronwall's inequality applied to the difference, so $m=m^*$. No strict convexity of the potentials was needed. This proposition is conditional on identifying a classical MFG limit; it is not such an identification theorem for all finite-game limits. ∎
Theorem (Full selection in the affine-potential subclass): Suppose $\mathcal F(m)=c\cdot m+C_0$ and $\mathcal G(m)=\gamma\cdot m+C_1$. Every sequence of equilibria from the input has empirical paths converging uniformly in probability to the unique global minimizing path $m^*$. In particular every subsequential law limit is $\delta_{m^*}$ and satisfies the requested selection property.
Proof: Here the individual interaction costs are $f_i=c_i$ and $g_i=\gamma_i$, independent of all other players' states. Solve the backward ODE \[ \dot u_i=H_i(u)-c_i,\qquad u(T)=\gamma, \qquad a^*_{ij}(t)=(u_i(t)-u_j(t))_+. \] Local existence and uniqueness hold because $H$ is locally Lipschitz. In reversed time $v(\tau)=u(T-\tau)$ the equation is $v_i'=c_i-H_i(v)$. A maximal component has derivative at most $\|c\|_\infty$; a minimal component has $H_i=0$ and derivative at least $-\|c\|_\infty$. These assertions also hold as one-sided derivatives at ties, by choosing an active extremal index. Equivalently, comparison with the scalar barriers $\pm(\|\gamma\|_\infty+\tau\|c\|_\infty)$ gives $\|u(t)\|_\infty\leq M:=\|\gamma\|_\infty+T\|c\|_\infty$. This prevents finite-time escape and extends the ODE through the whole horizon. Its feedback rates are continuous and at most $2M$.

For a single controlled state process $X$, possibly with rates depending on all players' observed histories, apply Dynkin's formula to $u_i(t)$ at $i=X_t$. For every bounded admissible control its expected cost equals \[ u_{X_0}(0)+\mathbb E\int_0^T\sum_{j\ne X_t} \left[\tfrac12(a_{X_tj}-a^*_{X_tj})^2+ (u_j-u_{X_t})_+a_{X_tj}\right]dt. \] This is the same scalar square-completion as in the preceding identity; the ODE and terminal condition cancel the other terms. Thus $a^*$ attains the minimum against any behavior of the other players. A Nash player's actual cost must attain this minimum, since it may deviate to $a^*$. The nonnegative square remainder then vanishes almost everywhere with respect to time and its realized law, forcing its actual outgoing intensities to equal $a^*$ on that law.

Consequently the joint state process has, up to null time--path sets, the deterministic generator equal to the sum of the independent single-particle generators with rates $a^*(t)$. A finite-state time-inhomogeneous generator with bounded continuous coefficients determines its transition matrices uniquely by the linear Kolmogorov equations. Their product transition law is a solution, hence is the unique joint law. The initial distribution is a product as well. This justifies independence of the players' realized states in this subclass, even if an equilibrium feedback was written using irrelevant other-player coordinates.

Let $m^*$ solve the corresponding linear forward equation from $m_0$. In the empirical decomposition the drift is the linear map $b(\mu^N,a^*)$. Its Lipschitz constant is bounded by some $L$ depending only on $d,M$. The coordinate martingale brackets are at most a constant times $T/N$, because each empirical jump has size $1/N$ and the total jump intensity is at most $2NM(d-1)$. Doob's inequality, the initial bound $\mathbb E\|\mu^N(0)-m_0\|_2^2\leq1/N$, and the pathwise integral inequality give \[ \sup_{t\leq T}\|\mu^N(t)-m^*(t)\|_2 \leq e^{LT}\left(\|\mu^N(0)-m_0\|_2+ \sup_{t\leq T}\|M^N(t)\|_2\right), \] \[ \mathbb E\sup_{t\leq T}\|\mu^N(t)-m^*(t)\|_2^2\leq\frac{C_{d,T,c,\gamma}}N. \] This proves uniform convergence in probability. Finally affine potentials have zero Bregman remainders, and $(m^*,u)$ satisfies the classical MFG equations. The verification identity proves global optimality and uniqueness of its minimizing population path. The argument applies to every specified equilibrium, not only a selected favorable one. ∎

Proposition (An explicit nonminimizing MFG and a failed finite-game candidate): For $d=3$, $T=1$, $m_0=(1/3,1/3,1/3)$, $\mathcal F=0$ and $\mathcal G(m)=-3(m_1-1/3)^2$, the constant zero-control MFG path is not a potential minimizer. The profile prescribing zero rates in every finite-player configuration is not Markov perfect for any $N\geq2$.
Proof: The constant law and values $u=0$ satisfy the deterministic backward--forward equations because $g(m_0)=0$. Its objective is zero. The control $a_{21}=a_{31}=\log2$, all other rates zero, gives $m(t)=(1-\frac23e^{-t\log2},\frac13e^{-t\log2},\frac13e^{-t\log2})$ and $J=(\log2)/6-1/3<0$. This verifies nonminimality with an explicit finite-energy competitor.

To check the finite game, take a configuration where all players other than $\ell$ are in state one and player $\ell$ is in state two. Under zero feedbacks its continuation cost is zero. Its opponents remain in state one, so its own terminal costs are $g_1=-4,g_2=g_3=0$. Deviate by jumping from state two to one at constant rate $c\in(0,8)$ and making no further jumps. Over a remaining horizon $r>0$ this has expected cost \[ \left(\tfrac c2-4\right)(1-e^{-cr})<0. \] The energy term is $\frac12c^2\int_0^r e^{-ct}dt$, and the terminal term is $-4(1-e^{-cr})$, yielding the formula. The deviation is bounded and admissible. Thus zero feedback fails the required continuation Nash inequality. In fact this configuration has positive probability under the prescribed independent interior initial law, however small that probability becomes as $N$ grows. ∎

Proposition (A complete counterexample for the ex ante approximate-Nash variant): In the same three-state example, the zero-feedback profiles are ex ante $\eta_N$-Nash equilibria with $\eta_N=2\sqrt{2/(N-1)}\to0$. Their empirical laws converge to the nonminimizing constant MFG path. Thus replacing exact Markov perfection by ex ante vanishing-error Nash would make the selection assertion false.
Proof: Fix a player. When all other players have zero rates, their initial empirical fraction $Y$ in state one is constant in time, with mean $1/3$ and variance $2/[9(N-1)]$. Its own terminal cost in state one is $g_1(Y)=-6(Y-1/3)$ and in the other states it is zero. For any bounded progressively measurable deviation, kinetic cost is nonnegative and terminal cost is at least $-|g_1(Y)|$. Therefore its deviating expected cost is at least \[ -\mathbb E|g_1(Y)|\geq-6\sqrt{\operatorname{Var}(Y)} =-2\sqrt{2/(N-1)}. \] Under zero rates, its initial state is independent of $Y$, so its expected cost is $(1/3)\mathbb E g_1(Y)=0$. This bounds every ex ante unilateral improvement by $\eta_N$. Every empirical path is its initial empirical law held constant, which converges in probability to $m_0$. Nonminimality was proved above. This approximate result does not hold uniformly over continuation configurations, as the preceding deviation with $g_1=-4$ demonstrates. ∎
Computation.
The verification script checks the explicit competitor's terminal law $(2/3,1/6,1/6)$ and objective approximately $-0.2178088032$. The square-completion identity is exact algebra, and the finite-game deviation inequality follows from $0<c<8$, not a numerical optimization. Minimal pseudocode for testing the proposed zero finite-game feedback is: 
\begin{lstlisting} for N in [2, 3, 4]:     # Other N-1 players all occupy state 1; deviator starts in state 2.     for c in [1, 2, 4]:         deviation_cost = (c/2 - 4)*(1-exp(-c))         assert deviation_cost < 0 \end{lstlisting}
The supplied script evaluates the same analytic construction; neither these cases nor a simulation is used to assert an exact-Nash counterexample.

5) VERIFICATION

The flux formulation proves that the event $\mathcal M_{T,m_0}$ is a Borel event, avoiding a hidden measurability issue in $Q(\mathcal M)=1$. The compactness theorem uses an actual allowed zero-rate deviation and independent jump compensators, not a uniform rate bound absent from the input. The verification identity includes the extra term $(u_j-u_i)_+a_{ij}$; omitting it when $u_i<u_j$ would make the square completion false. Finite energy suffices for all integrations by parts. The affine theorem covers all equilibria up to irrelevant null-set prescriptions. The nonconvex candidate was explicitly tested at rare off-path configurations and fails exact Markov perfection. Its valid approximate-Nash counterexample is labeled with that weaker, ex ante quantifier only.

\Needspace{8\baselineskip}

6) FINAL

LABEL: UNRESOLVED for the exact Markov-perfect selection question with arbitrary smooth nonconvex potentials.

Strongest checked partial results. General compactness of exact-equilibrium empirical laws with continuous limits; compactness of the global minimizing-path set; an exact potential verification identity; full selection for every equilibrium in the affine-potential subclass; and a complete counterexample to the weaker ex ante approximate-Nash formulation.

Exact first gap. Exact finite-player Nash inequalities have not been converted into a global potential-minimization inequality for every limiting population path when the potential Bregman remainders can be negative.

Top three next moves. (1) Quantify the discrepancy between the finite-game Nash system and a discrete global potential optimization problem, including all $m^{N,-\ell}$ shifts. (2) Identify limit controls using the proved energy compactness while retaining the continuation optimality information, not merely ex ante approximate optimality. (3) Test the explicit concave terminal potential using actual finite-$N$ Markov-perfect feedbacks rather than the rejected zero-rate candidate.

Minimal counterexample perspective. A candidate must retain a nonminimizing path while remaining an exact best response at every finite configuration, including exponentially rare ones. The displayed zero-rate profiles show why merely approximating a deterministic nonminimizing equilibrium is insufficient.

Reproducibility and scope.
This is one of eleven companion reports. The bundle includes verify.py and its executed results. Finite experiments supplement the proofs; they are not proofs of asymptotic claims. Literature coverage is targeted, and no novelty or formal proof-assistant certification is claimed.

References

  • [cardaliaguet-delarue] Pierre Cardaliaguet and François Delarue. Selected topics in mean field games. Proceedings of the International Congress of Mathematicians 2022, volume 5. DOI: 10.4171/ICM2022/17. Sections 4.3--4.4; especially printed pp. 3691--3693. https://ems.press/content/book-chapter-files/33265.

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Econ-28-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-28

An action-gap test for finite-state potential selection

Problem C73-27. Status: unresolved. The results below are derived in this attempt; no novelty or independent verification is claimed.

The target and the restricted result

The target concerns limits of exact symmetric Markov-perfect equilibria of finite games with $d\ge3$ states. It asks whether their population paths globally minimize the deterministic potential action. The distinction between a first-order equilibrium and a global minimizer is essential [CD, Section 4.3]. We prove an explicit curvature--horizon condition under which every classical deterministic equilibrium is a global minimizer, and construct a nonminimizing equilibrium outside that condition. Neither result identifies arbitrary finite-player limits.

Let $\Delta_d=\{m\ge0:\sum_i m_i=1\}$, let $\mathcal F,\mathcal G$ be $C^2$ on a neighborhood of $\Delta_d$, and write $f_i=\partial_i\mathcal F$, $g_i=\partial_i\mathcal G$. For measurable nonnegative off-diagonal rates $a_{ij}$ and an absolutely continuous population path $m$, admissibility means \[ \dot m_i=\sum_{j\ne i}(m_ja_{ji}-m_i a_{ij}),\quad m(0)=m_0, \qquad\int_0^T\sum_{i,j\ne i}m_i a_{ij}^2{\mathrm d} t<\infty. \] The objective is \[ J(m,a)=\int_0^T\left[\frac12\sum_{i,j\ne i}m_i a_{ij}^2+\mathcal F(m)\right]{\mathrm d} t+\mathcal G(m(T)). \] A classical equilibrium is a continuously differentiable pair $(m,u)$ with \begin{equation}\label{wsref-0-65-71-75-69-6c} -\dot u_i+H_i(u)=f_i(m),\quad H_i(u)=\frac12\sum_{j\ne i}(u_i-u_j)_+^2,\quad a^*_{ij}=(u_i-u_j)_+,\quad u_i(T)=g_i(m(T)), \end{equation} and the forward equation for $m,a^*$. Such rates are bounded on the fixed time interval.

An exact gap decomposition

For $\mathcal K\in\{\mathcal F,\mathcal G\}$ set $B_{\mathcal K}(n,m)=\mathcal K(n)-\mathcal K(m)-D\mathcal K(m)\cdot(n-m)$.
Lemma: For every equilibrium $(m,u,a^*)$ and every admissible competitor $(n,a)$ with the same initial law, \begin{align}\label{wsref-0-67-61-70-69-64} J(n,a)-J(m,a^*)={}&\int_0^T\sum_{i,j\ne i}n_i\left[ \frac12(a_{ij}-a^*_{ij})^2+a_{ij}(u_j-u_i)_+\right]{\mathrm d} t\\ &+\int_0^T B_{\mathcal F}(n,m){\mathrm d} t+B_{\mathcal G}(n(T),m(T)).\nonumber \end{align} All terms are finite except possibly the nonnegative squared-rate term; for finite-energy admissible competitors that term is finite too.
Proof: Let $\delta=n-m$. The forward equations give \[ \frac{\mathrm d}{{\mathrm d} t}(u\cdot\delta) =\dot u\cdot\delta+\sum_{i,j\ne i}[n_i a_{ij}-m_i a^*_{ij}](u_j-u_i). \] Integrate this relation, use $\delta(0)=0$, $u(T)=D\mathcal G(m(T))$, and $f_i(m)=H_i(u)-\dot u_i$. The linear parts of the two potential differences then combine with the kinetic cost difference to give \[ \int_0^T\sum_i n_i\sum_{j\ne i} \left[\tfrac12a_{ij}^2+a_{ij}(u_j-u_i)+\tfrac12(u_i-u_j)_+^2\right]{\mathrm d} t. \] The corresponding expression at $(m,a^*)$ is zero, because $a^*_{ij}$ minimizes $a^2/2+a(u_j-u_i)$ over $a\ge0$. For real $q$ and $a\ge0$, \[ \tfrac12a^2-aq+\tfrac12q_+^2 =\tfrac12(a-q_+)^2+a(-q)_+. \] Applying this identity with $q=u_i-u_j$ proves (gapid). All integrations are legitimate: boundedness of $u,a^*$ and Cauchy--Schwarz with $\sum n_i=1$ control the linear rate terms by the finite energy. ∎
Lemma (Population displacement controlled by rate error): With the same notation define \[ e(t)=\sum_{i,j\ne i}n_i(t)(a_{ij}(t)-a^*_{ij}(t))^2,\qquad E=\int_0^T e(t){\mathrm d} t. \] Then \[ \|n(t)-m(t)\|_2^2\le4(d-1)t\int_0^t e(s){\mathrm d} s, \quad \int_0^T\|n-m\|_2^2{\mathrm d} t\le2(d-1)T^2E. \]
Proof: Let $A^*(t)$ be the rate matrix with off-diagonal entries $a^*_{ij}$. The difference solves $\dot\delta=(A^*)^\mathsf{T}\delta+h$, where $h_i=\sum_{j\ne i}[n_j(a_{ji}-a^*_{ji})-n_i(a_{ij}-a^*_{ij})]$. The fundamental solution of $(A^*)^\mathsf{T}$ is a stochastic matrix acting on column laws, hence is a contraction in $\ell^1$: positivity and column sums one give this directly by the triangle inequality. Variation of constants implies \[ \|\delta(t)\|_1\le\int_0^t\|h(s)\|_1{\mathrm d} s \le2\sqrt{d-1}\int_0^t\sqrt{e(s)}{\mathrm d} s. \] Here Cauchy--Schwarz uses $\sum_{i,j\ne i}n_i=d-1$. Square, use time Cauchy--Schwarz and $\|\delta\|_2\le\|\delta\|_1$. Integrating the first bound and interchanging nonnegative integrals gives $4(d-1)\int_0^T e(s)\int_s^T t{\mathrm d} t{\mathrm d} s\le2(d-1)T^2E$. ∎
Theorem (A global minimum certificate): Suppose for every $m,n\in\Delta_d$, \[ B_{\mathcal F}(n,m)\ge-\frac L2\|n-m\|_2^2,\qquad B_{\mathcal G}(n,m)\ge-\frac {L_G}2\|n-m\|_2^2 \] with $L,L_G\ge0$. Define \[ \eta=\frac12-(d-1)LT^2-2(d-1)L_GT. \] If $\eta\ge0$, every classical equilibrium is a global minimizer of $J$. If $\eta>0$ and one classical equilibrium exists, its law is the unique minimizing law path; any other classical equilibrium has that same law. More precisely, every admissible competitor satisfies \[ J(n,a)-J(m,a^*)\ge\eta\int_0^T\sum_{i,j\ne i}n_i(a_{ij}-a^*_{ij})^2{\mathrm d} t. \]
Proof: Drop the nonnegative inactive-edge term in Lemma Lemma. Apply the two semiconvexity inequalities and Lemma Lemma (Population displacement controlled by rate error), including its terminal-time bound. This gives $J(n,a)-J(m,a^*)\ge[1/2-(d-1)LT^2-2(d-1)L_GT]E$. For $\eta\ge0$ this proves minimality. If $\eta>0$ and equality holds, then $E=0$, and the first estimate of Lemma Lemma (Population displacement controlled by rate error) gives $n=m$ at every time. Every classical equilibrium is itself minimizing by the first conclusion, so its law agrees as well. Rates on zero-mass states need not be unique, which is why uniqueness is asserted for laws. ∎

The criterion includes nonconvex potentials and has the expected running-cost curvature scale $LT^2$. No strict monotonicity is assumed. For convex potentials $L=L_G=0$, the identity yields minimality at every horizon, but it remains a statement about deterministic equilibria.

An explicit nonminimizing equilibrium

Proposition: Fix $d\ge3$, $m_0=(1/d,\ldots,1/d)$, and \[ \mathcal F(m)=-\frac K2\|m-m_0\|_2^2,\qquad\mathcal G=0. \] For every $T>0$, $m(t)=m_0$, $u(t)=0$, $a^*(t)=0$ is a classical equilibrium with action zero. If $K>3d/(2T^2)$, it is not a global minimizer.
Proof: At $m_0$, $D\mathcal F=0$, so all equations (equil) hold. To produce a competitor, choose only the rate $a_{12}=h>0$ and set all others to zero. Then \[ n_1(t)=d^{-1}e^{-ht},\quad n_2(t)=d^{-1}+d^{-1}(1-e^{-ht}),\quad n_i(t)=d^{-1}\ (i\ge3). \] This is admissible and has action \[ J(n,a)=\frac{h}{2d}(1-e^{-hT})-\frac K{d^2}\int_0^T(1-e^{-ht})^2{\mathrm d} t. \] Divide by $h^2$ and let $h\downarrow0$. The first term tends to $T/(2d)$ and the second to $-KT^3/(3d^2)$, by dominated convergence using $(1-e^{-ht})/h\le t$. Their sum is negative under the stated inequality, so $J(n,a)<0$ for all sufficiently small positive $h$. ∎

This is not a counterexample to the finite-player selection conjecture: no assertion is made that these zero-control paths arise as limits of exact Markov-perfect finite-game equilibria. It rules out the shortcut "every solution of the deterministic forward--backward equations is minimizing."

Connection to finite-player limits and the remaining step

Corollary (Conditional support statement): Assume $\eta\ge0$ in Theorem Theorem (A global minimum certificate). Let $Q$ be any probability law on continuous population paths such that $Q$-almost every path can be completed to a classical deterministic equilibrium with initial law $m_0$. Then $Q$ is supported on $\mathcal M_{T,m_0}$. If $\eta>0$ and such paths exist, $Q$ is a Dirac mass at the unique minimizing law.
Proof: Apply Theorem Theorem (A global minimum certificate) path by path. For the strict case, its uniqueness conclusion identifies all these paths with the same minimizing law, proving the Dirac assertion. ∎

The support assumption is additional. Randomized limits of closed-loop finite games must be identified with the correct limiting equilibrium concept; conditioning a weak random equilibrium on its entire population path need not preserve its information structure. We have not proved that arbitrary subsequences in the catalogue have the support property used in the corollary. The first unresolved step is precisely that identification, followed outside the curvature regime by a mechanism excluding the nonminimizing equilibria exemplified above. The source's potential-noise selection theorem uses a different regularization [CeD]; it is not a large-$N$ theorem for the original games.

References

  • [CD] P. Cardaliaguet and F. Delarue, Selected topics in mean field games, Proc. ICM 2022, vol. 5, 3660--3703, EMS Press (2023), Section 4.3, Proposition 4.4 and the following selection discussion. https://doi.org/10.4171/ICM2022/17.
  • [CeD] A. Cecchin and F. Delarue, Selection by vanishing common noise for potential finite state mean field games, Comm. Partial Differential Equations 47 (2022), 89--168. Theorem 2.7 and Section 2.6 in the preprint explain the potential structure and the distinct finite-player issue. https://arxiv.org/abs/2005.12153.

Completion Estimate

10%

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Econ-28-A3 · Full disproof claimed · LLM-generated · gpt-6 astra, claude opus 5.5, gpt-6 astra pro

C73-27: exact finite-player selection of a nonminimizing potential equilibrium

C73-27: exact finite-player selection
of a nonminimizing potential equilibrium

Ji Ho Bae
LawAll AI

10 October 2026

\begin{abstract} Fixed smooth three-state data give bounded symmetric exact Markov-perfect feedbacks for every population size. Their empirical paths converge to a constant equilibrium that does not minimize the potential action. We verify every continuation configuration and all bounded progressively measurable unilateral deviations, prove the full path limit, and exhibit a strictly better feasible path. \end{abstract}

Statement and smooth data

Write $\Delta_3=\{m\in[0,1]^3:\sum_i m_i=1\}$ and fix $T=1$, $m_0=(1/3,1/3,1/3)$. For the potential action in C73-27, \begin{equation}\label{wsref-0-61-63-74-69-6f-6e} J(m,a)=\int_0^1\left[\frac12\sum_{i,j\ne i}m_i a_{ij}^2+ \mathcal F(m)\right]dt+\mathcal G(m(1)),\qquad \dot m_i=\sum_{j\ne i}(m_ja_{ji}-m_ia_{ij}), \end{equation} the admissible paths are absolutely continuous simplex paths starting at $m_0$ with measurable nonnegative off-diagonal rates and finite weighted quadratic energy. Denote the paths in global minimizers by $\mathcal M_{1,m_0}$. In the finite game, player $\ell$'s terminal cost is $g_{X_1^\ell}(m_1^{N,-\ell})$, where $g_i=\partial_i\mathcal G$ and $m^{N,-\ell}$ is the empirical law of the other $N-1$ players. The running cost is $\frac12\sum_{j\ne X_t^\ell}(a^\ell_{X_t^\ell j})^2+ f_{X_t^\ell}(m_t^{N,-\ell})$, with $f_i=\partial_i\mathcal F$.
Theorem: There are $C^\infty$ potentials on $\mathbb R^3$ and, for every $N\ge2$, bounded symmetric Markov-perfect feedbacks for exactly these games whose empirical processes satisfy \[ \sup_{0\le t\le1}\|\mu_t^N-m_0\|_2\longrightarrow0 \quad\text{in probability},\qquad \mathcal L(\mu^N)\Longrightarrow Q=\delta_{\bar m} \quad\text{in }D([0,1],\Delta_3)\text{ with }J_1, \] where $\bar m(t)=m_0$. This continuous limit has $Q(\mathcal M_{1,m_0})=0$. In particular, the universal selection assertion in the stated C73-27 formulation is false.

Set $a=2/5$, $b=3/5$ and define \[ \rho(r)=\begin{cases}e^{-1/r},&r>0,\\0,&r\le0,\end{cases} \qquad \psi(z)=\frac{\rho(z-a)}{\rho(z-a)+\rho(b-z)}. \] The denominator is positive everywhere. Every right derivative of $e^{-1/r}$ is a polynomial in $1/r$ times $e^{-1/r}$ and tends to zero at $0$, so $\rho$ and $\psi$ are smooth. The quotient rule gives \[ \psi'(z)=\frac{\rho'(z-a)\rho(b-z)+\rho(z-a)\rho'(b-z)} {[\rho(z-a)+\rho(b-z)]^2}\ge0. \] Thus $\psi=0$ on $(-\infty,a]$, $\psi=1$ on $[b,\infty)$, and $L:=\sup_\mathbb R\psi'<\infty$. With $S(m)=\sum_i m_i$, choose \begin{equation}\label{wsref-0-64-61-74-61} \mathcal F(m)=0,\qquad \mathcal G(m)=-\psi(m_1)+(S(m)-1)\psi'(m_1). \end{equation} These are fixed independently of $N$. Direct ambient differentiation yields $\partial_1\mathcal G=(S-1)\psi''(m_1)$ and $\partial_2\mathcal G=\partial_3\mathcal G=\psi'(m_1)$. Consequently on the simplex the actual data are \begin{equation}\label{wsref-0-63-6f-73-74-73} \mathcal G(m)=-\psi(m_1),\qquad (g_1,g_2,g_3)=(0,\psi'(m_1),\psi'(m_1)),\qquad f_i=0. \end{equation} All individual terminal costs lie in $[0,L]$.

An exact feedback for every finite game

For $x\in\{1,2,3\}^N$ let $k_\ell(x)=\#\{r\ne\ell:x_r=1\}$ and write $x^{\ell\to j}$ for the vector with coordinate $\ell$ replaced by $j$. Construct scalar functions in reverse time $\tau\in[0,1]$ by \begin{align} w_{N-1}'&=-\tfrac12w_{N-1}^2,& w_{N-1}(0)&=\psi'(1)=0,\label{wsref-0-6f-64-65-2d-74-6f-70}\\ w_k'&=(N-1-k)w_k w_{k+1}-(N-k-\tfrac12)w_k^2, &w_k(0)&=\psi'\left(k/(N-1)\right),\quad 0\le k<N-1. \label{wsref-0-6f-64-65} \end{align}
Lemma: This system has a unique $C^1$ solution with $0\le w_k\le L$. If $\psi'(k/(N-1))=0$, then $w_k\equiv0$.
Proof: The top function is zero. Proceed downwards, assuming $0\le w_{k+1}\le L$. Put $A_k=(N-1-k)w_{k+1}$, $B_k=N-k-1/2$ and $c_k=w_k(0)$. The equation $w_k'=A_kw_k-B_kw_k^2$ is locally Lipschitz in $w_k$. For $c_k=0$ its unique solution is zero. For $c_k>0$ its global solution is \[ w_k(\tau)=\frac{c_k\exp(\int_0^\tau A_k(r)dr)} {1+B_kc_k\int_0^\tau\exp(\int_0^s A_k(r)dr)ds}. \] Differentiation verifies the equation and initial value. The denominator is positive. Whenever $w_k>L$, its derivative is bounded above by $(N-1-k)Lw_k-(N-k-1/2)w_k^2<0$. A connected interval above $L$ would therefore be strictly decreasing from its entry value $L$, a contradiction. This proves the bound and the induction. ∎

Define values and outgoing feedbacks by \begin{equation}\label{wsref-0-76-61-6c-75-65-73} \begin{aligned} v_\ell(t,x)=\begin{cases}0,&x_\ell=1,\\ w_{k_\ell(x)}(1-t),&x_\ell\in\{2,3\},\end{cases} \\[3pt] q_j^\ell(t,x)=\begin{cases} w_{k_\ell(x)}(1-t),&x_\ell\in\{2,3\},\ j=1,\\ 0,&\text{otherwise}, \end{cases}\quad j\ne x_\ell. \end{aligned} \end{equation} The diagonal is minus the outgoing sum. Rows for counterfactual own states are given by the same rule after replacing $x_\ell$ by that state, so these define admissible full rate matrices. They are continuous in time, bounded by $L$ independently of $N$, and symmetric under player permutations. Independent Poisson transition noises implement the joint chain; its total intensity is at most $NL$, so it is nonexplosive and uses no common noise.

The exact Nash equations are \begin{equation}\label{wsref-0-68-6a-62} \partial_t v_\ell+ \sum_{r\ne\ell}\sum_{j\ne x_r}q_j^r [v_\ell(t,x^{r\to j})-v_\ell(t,x)] -\tfrac12\sum_{j\ne x_\ell} [v_\ell(t,x)-v_\ell(t,x^{\ell\to j})]_+^2=0. \end{equation} Their terminal condition is exactly $v_\ell(1,x)=g_{x_\ell}(m^{N,-\ell}_x)$. To verify (hjb) at every configuration, first suppose $x_\ell=1$. Its value is zero under all opponent changes, and own alternative values are nonnegative. Thus all terms vanish and its optimal rates are zero. If $x_\ell\ne1$, set $k=k_\ell(x)$ and $\tau=1-t$. Its own jump to $1$ changes its value from $w_k$ to $0$; switching between $2$ and $3$ leaves its value unchanged. Its own Hamiltonian is therefore $w_k^2/2$. Of the opponents, the $k$ in state $1$ have zero rates. Each of the $N-1-k$ others has exactly $k$ state-$1$ opponents and jumps to $1$ at rate $w_k$, changing the focal value to $w_{k+1}$. Equation (hjb) becomes $-w_k'+(N-1-k)w_k(w_{k+1}-w_k)-w_k^2/2=0$, precisely (ode); when $k=N-1$ it is (ode-top). In all cases, \begin{equation}\label{wsref-0-6f-70-74-69-6d-61-6c-2d-72-61-74-65} q_j^\ell(t,x)=[v_\ell(t,x)-v_\ell(t,x^{\ell\to j})]_+. \end{equation}

Proposition: The feedbacks (values) are exact Markov-perfect Nash equilibria against every bounded progressively measurable unilateral rate deviation.
Proof: Fix any initial continuation time $t_0$, state vector $x$ and player $\ell$, retaining the opponents' feedbacks. Let its outgoing rates $\alpha_j(t)\ge0$ be any admissible bounded progressively measurable deviation, possibly depending on the entire history, and let $X$ be the resulting state process. Put $\Delta_j=v_\ell(t,X_t^{\ell\to j})-v_\ell(t,X_t)$. The compensated-jump formula has a mean-zero martingale because the values, their time derivatives for fixed $N$, and all intensities are bounded. Using (hjb) in that formula gives exactly \begin{align}\label{wsref-0-76-65-72-69-66-79} &\mathbb E\left[\int_{t_0}^1\tfrac12\sum_{j\ne X_t^\ell}\alpha_j(t)^2dt +g_{X_1^\ell}(m_1^{N,-\ell})\right]-v_\ell(t_0,x)\nonumber\\ &\quad=\mathbb E\int_{t_0}^1\sum_{j\ne X_t^\ell} \left[\tfrac12(\alpha_j-q_j^\ell)^2+\alpha_j(\Delta_j)_+\right]dt\ge0. \end{align} Indeed, for $\alpha\ge0$ and $q=(-\Delta)_+$, $\alpha^2/2+\alpha\Delta+q^2/2 =(\alpha-q)^2/2+\alpha\Delta_+$. By (optimal-rate) the prescribed feedback makes this remainder zero. It therefore attains the minimum against all allowed deviations. The choices of $t_0,x,\ell$ were arbitrary, proving the continuation condition at every time and configuration, including rare states. ∎

Convergence of the entire empirical path

Take the required independent initial states with law $m_0$, independent of the transition noises. Let $K_N$ count those initially in state $1$, and set $A_N=\{K_N\le(2/5)(N-1)\}$. On $A_N$, every player outside state $1$ has $K_N$ state-$1$ opponents, so its initial scalar terminal datum is zero. Lemma Lemma gives $w_{K_N}(1-t)=0$ at every time. Players in state $1$ also have zero rates. This configuration is absorbing under the exact feedback, and $\mu_t^N=\mu_0^N$ throughout $[0,1]$ on $A_N$.

Here $K_N$ is binomial with mean $N/3$ and variance $2N/9$. For $N\ge12$ the threshold exceeds its mean by at least $N/30$; Chebyshev's inequality gives $\mathbb P(A_N^c)\le200/N$. Also $\mathbb E\|\mu_0^N-m_0\|_2^2=2/(3N)$. Since the squared simplex diameter is at most $2$, \begin{equation}\label{wsref-0-63-6f-6e-76-65-72-67-65-6e-63-65} \mathbb E\sup_{t\le1}\|\mu_t^N-m_0\|_2^2 \le\frac{2}{3N}+2\mathbb P(A_N^c) \le\frac{2}{3N}+\frac{400}{N}\longrightarrow0. \end{equation} Markov's inequality proves uniform convergence in probability. Choosing the identity time change bounds the $J_1$ distance by the uniform distance, so the laws converge in the $J_1$ topology to $Q=\delta_{\bar m}$. The limiting path is continuous.

Existence of minimizers and strict nonminimality

For completeness, the minimizing-path set is nonempty and Borel. Introduce fluxes $z_{ij}=m_i a_{ij}\ge0$. The constraints become linear in $z$, and the energy is $K(m,z)=\frac12\int_0^1\sum z_{ij}^2/m_i\,dt$, with $0/0=0$ and $z^2/0=+\infty$ for $z>0$. The constant zero-flux path is feasible. Since the terminal potential belongs to $[-1,0]$, a minimizing sequence has bounded energy, hence bounded $L^2$ fluxes because $m_i\le1$. The forward equation gives bounded $L^2$ path derivatives and a common $1/2$-Hölder modulus. Arzelà--Ascoli and weak $L^2$ compactness give uniform $m^n\to m$ and weak $z^n\rightharpoonup z$ along a subsequence. Nonnegative test functions preserve $z\ge0$, and time interval indicators pass the integrated constraints to the limit.

The exact dual representation \[ K(m,z)=\sup_{q\in L^\infty}\int_0^1\sum_{i,j\ne i} [q_{ij}z_{ij}-\tfrac12m_iq_{ij}^2]dt \] proves lower semicontinuity: each bounded $q$ gives a continuous expression under these convergences. Equality in the representation follows by the bounded choices $q_{ij}=\min\{A,z_{ij}/m_i\}$, assigning $A$ where $m_i=0<z_{ij}$ and zero where both vanish, and monotone convergence as $A\uparrow\infty$. Continuity passes the terminal potential to the limit. Thus the limit minimizes. Finite energy forces $z_{ij}=0$ almost everywhere on $\{m_i=0\}$; defining $a_{ij}=z_{ij}/m_i$ on $\{m_i>0\}$ and zero elsewhere recovers admissible measurable rates. The same argument for any sequence of minimizers shows that their law paths form a uniformly compact set. Its inclusion in $D$ is continuous for $J_1$, so it is compact and Borel there as well.

Every rate representation of $\bar m$ has nonnegative action, since $\mathcal F=0$, $\mathcal G(m_0)=0$, and kinetic energy is nonnegative. For a competitor choose $c=\log2$, $a_{21}=a_{31}=c$, and all other off-diagonal rates zero. The forward law is \[ m_1(t)=1-\tfrac23e^{-ct},\qquad m_2(t)=m_3(t)=\tfrac13e^{-ct}. \] It is admissible and has terminal law $(2/3,1/6,1/6)$, where $\mathcal G=-1$. Its energy is $\frac12\int_0^1(m_2+m_3)c^2dt=c(1-e^{-c})/3=(\log2)/6$. Therefore \begin{equation}\label{wsref-0-73-74-72-69-63-74} J(m,a)=\frac{\log2}{6}-1<0. \end{equation} For example $\log2=\int_1^2x^{-1}dx<1$ proves this strict inequality. The global minimum is negative, so $\bar m$ cannot occur in any minimizing pair. This gives $Q(\mathcal M_{1,m_0})=0$ and completes Theorem Theorem.

The prescribed representatives and source scope

The normal term in (data) vanishes on the simplex but adds the common component $\psi'(m_1)$ to the ambient gradient of $-\psi(m_1)$. Its tangent projection is $(-2\psi'/3,\psi'/3,\psi'/3)$, exactly the intrinsic gradient of the restricted potential. This is allowed by the ambient derivative prescription in C73-27 and by the projected potential convention in [CeD, Eq. $2.9$]; no hypothesis requires $\sum_i g_i=0$. We retain the actual costs (costs) in every finite continuation game. State-dependent common additions are not asserted to preserve finite closed-loop feedbacks. Centering those costs would change the finite game and would require another analysis.

The published two-state selection result analyzes a specified anti-monotone model, rather than all smooth terminal costs: [CDFP, Section 3] uses $G(x,m)=-mx$ on $\{-1,1\}$ with zero interaction running cost. The potential selection discussion in [ICM, Section 4.3] supplies the motivation; the theorem here addresses the catalogue formulation.

Use of AI tools

The author used research AI agents that the author built and configured (GPT-6 Astra max, Claude Opus 5.5 max and GPT-6 Astra Pro) during the proof work, for proof search and construction, proof checking, translation, Python computation and similar tasks. The final manuscript was written, and the proofs were checked, by the human author. All responsibility for this work rests with the author personally.

References

  • [ICM] P. Cardaliaguet and F. Delarue, Selected topics in mean field games, Proc. ICM 2022, Vol. 5, EMS Press $2023$, 3660--3703, doi:10.4171/ICM2022/17, Section 4.3, equations $4.18$--$4.19$. https://ems.press/content/book-chapter-files/33265.
  • [CeD] A. Cecchin and F. Delarue, Selection by vanishing common noise for potential finite state mean field games, Comm. Partial Differential Equations 47 $2022$, 89--168, doi:10.1080/03605302.2021.1955256, equation $2.9$. https://arxiv.org/abs/2005.12153.
  • [CDFP] A. Cecchin, P. Dai Pra, M. Fischer and G. Pelino, On the convergence problem in mean field games: a two state model without uniqueness, arXiv:1810.05492v2 $2019$, Section 3. https://arxiv.org/abs/1810.05492.

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