应用度量正则性计算矩阵博弈平滑算法的条件测度

New estimates on the regularity of the pressure in density-constrained Mean Field Games

We consider variational Mean Field Games endowed with a constraint on the maximal density of the distribution of players. Minimizers of the variational formulation are equilibria for a game where both the running cost and the final cost of each player is augmented by a pressure effect, i.e. a positive cost concentrated on the set where the density saturates the constraint. Yet, this pressure is a priori only a measure and regularity is needed to give a precise meaning to its integral on trajectories. We improve, in the limited case where the Hamiltonian is quadratic, which allows to use optima…

Regularity of the minmax value and equilibria in multiplayer Blackwell games

A real-valued function $φ$ that is defined over all Borel sets of a topological space is \emph{regular} if for every Borel set $W$, $φ(W)$ is the supremum of $φ(C)$, over all closed sets $C$ that are contained in $W$, and the infimum of $φ(O)$, over all open sets $O$ that contain $W$. We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player's minmax value is regular. We then use the regularity of the minmax value to establish the existence of $\varep…

Zero-Sum Games involving Teams against Teams: Existence of Equilibria, and Comparison and Regularity in Information

Many emerging problems involve teams of agents taking part in a game. Such problems require a stochastic analysis with regard to the correlation structures among the agents belonging to a given team. In the context of Standard Borel spaces, this paper makes the following contributions for two teams of finitely many agents taking part in a zero-sum game: (i) An existence result will be presented for saddle-point equilibria in zero-sum games involving teams against teams when common randomness is assumed to be available in each team with an analysis on conditions for compactness of strategic tea…

无逐点等价条件下对应分布的正则性质：在大博弈中的应用

Regularity properties of distributions of correspondences without countable generation: applications to large games

本文证明对应的正则条件分布的四类正则性质均等价于无处等价条件，无需可数生成假设，并据此建立一般特征空间大博弈及半匿名情形下纯策略均衡的存在性。

Sobolev regularity for first order Mean Field Games

In this paper we obtain Sobolev estimates for weak solutions of first oder variational Mean Field Game systems with coupling terms that are local function of the density variable. Under some coercivity condition on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity condition on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open i…

Regularity and long time behavior of one-dimensional first-order mean field games and the planning problem

We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a non-separated, strictly monotone dependence on the density. We improve upon the existing literature by obtaining two regularity results. The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities. The second result is the interior smoothness of weak solutions without the need to assume neither blow-up of the cost function nor that the in…

Classical and weak solutions to local first-order mean field games through elliptic regularity

We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in the density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from…

停止者与奇异控制器博弈中值函数的全局正则性

Global regularity of the value function in a stopper vs. singular-controller game

研究停止者与奇异控制器零和博弈，证明值函数在全局为C^1，二阶导数除停止边界外连续，为分析自由边界和最优策略奠定基础。

多变量需求空间下路由博弈的无政府状态价格的正则性

Regularity of the Price of Anarchy in Routing Games over the Multivariate Demand Space

该研究分析多变量需求空间下路由博弈的无政府状态价格（PoA）的正则性，证明其在正则需求区域连续可微，不可微集测度为零，仿射成本下PoA最大值在单元边界取得，并在Sioux Falls网络验证理论。

非凸非光滑函数和的梯度映射型平稳性测度的渐近分析及在邻近梯度型算法中的应用

Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms

本文针对非凸非光滑函数和提出梯度映射型平稳性测度并做渐近分析，将其应用于邻近梯度算法，解决了一个开放问题并提出带非单调线搜索的邻近变量平滑算法。

Proving Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control

We reveal an interesting convex duality relationship between two problems: (a) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (b) a controller-and-stopper problem, in which the controller controls the drift and volatility of a process in order to maximize a running reward based on that process, and the stopper chooses the time to stop the running reward and rewards the controller a final amount at that time. Our primary goal is to show that the minimal probability of ruin, whose stoc…

Optimal Algorithms for Non-Smooth Distributed Optimization in Networks

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the er…

部分信息下的最优投资组合问题及具有相对绩效准则的均值场博弈

On the optimal portfolio problem with partial information and related mean field games with relative performance criteria

研究股票漂移部分信息下的最优投资组合选择，通过新方法求解单代理问题并推广到N人博弈的均值场极限，得到显式解和正则性结果。