23 problems

Mertens's conjectures. Every stochastic game with signals has a limit value. Moreover, when Player 1 is more informed than Player 2, the uniform maxmin and the limit value coincide…

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 $\lambda_{\rm max}:(0,1]^2\to[1,\infty)$ be the function measuring the maximum value associated with the game. Maximizer conjecture. The maximum value of…

Consider the multi-agent relative investment game in Section … can only be constant. The numerical experiments suggest convergence to a constant Nash equilibrium even when the suff…

Let $T_\infty=(h_1,h_2,\ldots)$ be a fixed partition with $h_1+h_2+\cdots=+\infty$, and let $v_{T_\infty,\lambda}(p)$ denote the value of the game from Theorem with discount parame…

An absorbing game has a limit value, and a rational absorbing game is one whose payoff and transition data are rational. A number is algebraic if it is a root of a nonzero polynomi…

Let $m\geq 1$. A rational absorbing game with $m$ actions per player is an absorbing game whose data are rational and in which each player has $m$ available actions. Maximal-degree…

Let samples be drawn from distributions that admit a density and have certain concentration properties. For a sample size $K_S$, let $\hat{P}_{K_S}$ denote the resulting empirical…

Let $\mathcal{S}=\mathbb{R}^{\mathcal{L}}_+$ be the set of nonnegative vectors indexed by locations $\mathcal{L}$. Consider a stochastic matching process for the two-level model an…

New conjecture. Under this perfectly observed state-component condition, Mertens's conjectures hold: the game has a limit value, and when Player 1 is more informed than Player 2, i…

Consider the stochastic multi-region SEIR model and the fictitious-play algorithm implemented with neural networks (NNs), whose parameters are initialized before the algorithm is r…

Given a pair $(\gamma,\Lambda)\in (\mathbb T^r,\mathbb L)$, let $\beta$ be the intensity of the account holder's randomized stopping strategy, with stopping intensity process…

Let $X^1,X^2,Y^1,Y^2,Z^1,Z^2$ be the payoff processes of a non-zero-sum Dynkin game, with payoff functions … … Define the regions…

Consider a filtered probability space with horizon $T$, adapted right-continuous payoff processes $X$, $Y$, and $Z$, and stopping times $\tau,\sigma$ in $\mathcal{T}_{[0,T]}$. Defi…

Let $\Phi^N(t)$ be the finite-dimensional matrix-valued function defined by the Riccati system in the source, with entries $\Phi^{N,i,j}(t)$. Consider the associated functions…

Consider the finite-player linear-quadratic stochastic differential game on a directed chain, its open-loop Nash equilibrium, and the analogous infinite-player game obtained as the…

Consider the periodic directed chain game and its finite-player open-loop Nash equilibrium, together with the corresponding infinite-player game obtained by taking the number of pl…

Let $N$ be the number of players, and let $n^N(t)$ denote the number of players in state $0$ at time $t$ under the unique symmetric Markov perfect equilibrium for the $N+1$-player…

Let a two-player zero-sum stochastic game have values that converge as the discount factor tends to $0$, with convergence uniform in the state space. For an initial state $k$, cons…

Let $v_\lambda$ denote the value vector associated with the parameter $\lambda>0$ in the stochastic game, and let $V$ be a finite set. Finite-limit-set conjecture. There exists a f…

Existence conjecture. The verification theorem can always be applied for the model considered; equivalently, for every situation to which the theorem applies, there exist character…

Let $N\in\mathbb{N}$ and let $I:=I(N)=\left[\frac{a}{\sqrt N},\frac{b}{\sqrt N}\right]$. For regular enough payoffs, let $\mathbb V:=\mathbb V(N)$ denote the corresponding super-re…

A second-order reflected backward stochastic differential equation with an upper obstacle would involve a decreasing process in its definition, in contrast to the increasing proces…