45 problems

Affine decoupling conjecture. According to the terminal condition of the filtering forward-backward stochastic differential equation, one has

Concavity conjecture. The function $F_\kappa$ is concave on $[0,\infty)$ for every $\beta,\sigma,\kappa>0$. The conjecture would imply that the synchronized stationary equilibrium,…

Single-agent robust mean field control limit conjecture. In the limit $N\to+\infty$, the problem should become a robust control problem involving a single agent, with a cost depend…

Graph-symmetry conjecture. Equilibrium impulse-response maps inherit the graph symmetry, reducing the fixed point to a single node coupled to its neighborhood and interpolating bet…

Consider the outer optimization iterates for the inverse mean field game problem, constrained by the discrete mean field game system, and let the feasible region denote the set of…

Let a potential mean field game be one for which the optimizers of an additional mean field control problem are also solutions to the mean field game, although some mean field game…

Controller's alpha-Rational Stackelberg equilibrium conjecture. This choice of initial action profile should imply that (\nu^*, i^{H}_*,\mu^R_*_{a^*_H}) is an $\alpha$-Rational S…

Consider the mean-field game in the supplied example, with state space consisting of $x=1,2$, population distribution $v\in\mathcal{P}(\mathcal{D})$, reward function $r$ independen…

Let $\Pi$ be the set of admissible stationary strategies, let $\pi^*\in\Pi$ be an equilibrium strategy, and let $\pi^i\otimes_1\pi^*$ denote the strategy that uses $\pi^i$ in the i…

Exponential ansatz for the value function. Based on the terminal condition, the value function is conjectured to have the form

Lower-bound conjecture.

Affine representation conjecture. One conjectures that

Let $b$ be a vector field satisfying Assumptions,, and. For each $N$ and configuration $\bm{x}$, let $\sigma_N(t,\bm{x})$ be the corresponding entropy solution, and let…

Let $w$ be the solution of the parabolic equation and let $(\lambda,\phi)$ be a viscosity solution of the ergodic HJB equation. Define … The bounded function $G$ should tend to a c…

The setting is cooperative multi-agent reinforcement learning with a global decision-making agent and homogeneous local agents, where the current method uses a generative oracle to…

Let $\bar{y}_i^{k-1}$ and $y_i^k$ denote the two possible initial guesses used to evaluate the best-response mapping $\textbf{BR}_{\lambda^k}(x_i)$ in the numerical algorithm, with…

Let $v^T(x,s):=u^T(x,sT)$ and $u^T(s):=m^T(sT)$ for $sin[0,1]$ and $xinmathbb{R}^n$, and let $c$ be the critical value in the ergodic problem. Write the first equation of that pro…

Let $(\lambda,c)$ and the Lagrangian $L$ be chosen so that there is a reversed monotonous mean-field-game travelling wave $({\mathcal V},{\Theta},{\mathcal M})$, and let…

Let $\Omega$ be the spatial domain in the mean field games system, and suppose that Lemmata 2.1 and 2.2 are established when $\Omega$ is a rectangular prism. Domain-extension conje…

Let the assumptions of Theorem … should remain true when \bm\mathfrak{R}^N and \overline\mathfrak{R} are replaced by \bm\mathcal{R}^N and \overline\mathcal{R}, respectively…

Let $N$ agents have exchangeable dynamics with weak coupling, and let $x^{(N)}$ and $u^{(N)}$ denote the relevant aggregate state and control quantities. Write $\mathbb{E}_0$ for e…

The Krusell–Smith growth model is a mean field game in which households are heterogeneous in wealth and productivity, and the optimal value depends on the distribution of agents' s…

Generic structural stability conjecture. For a generic 5-tuple $(f,L,\psi,\phi,\bar x)\in\mathcal{X}$, the mean field game has finitely many solutions, all of which are structurall…

For each interaction parameter $\kappa$, let $v^\gamma$ denote the value function associated with parameter $\gamma\geq 0$, and let $F_\kappa$ be the function defined in the paper…

Let $\kappa_c$ be the critical interaction parameter, and let $\kappa>\kappa_c$. A stationary solution is a probability measure on the torus whose constant flow solves the Kuramoto…