SIAM Journal on Control and Optimization, Volume 58, Issue 3, Page 1257-1288, January 2020.
We consider a class of ${N}$-player stochastic games of multidimensional singular control, in which each player faces a minimization problem of monotone-follower type with submodular costs. We call these games monotone-follower games. In a not necessarily Markovian setting, we establish the existence of Nash equilibria. Moreover, we introduce a sequence of approximating games by restricting, for each $n\in \mathbb{N}$, the players' admissible strategies to the set of Lipschitz processes with Lipschitz constant bounded by $n$. We prove that, for each $n\in \mathbb{N}$, there exists a Nash equilibrium of the approximating game and that the sequence of Nash equilibria converges, in the Meyer--Zheng sense, to a weak (distributional) Nash equilibrium of the original game of singular control. As a byproduct, such a convergence also provides approximation results of the equilibrium values across the two classes of games. We finally show how our results can be employed to prove existence of open-loop Nash equilibria in an $N$-player stochastic differential game with singular controls, and we propose an algorithm to determine a Nash equilibrium for the monotone-follower game.

中文翻译：

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非零和和次模单调跟随游戏：纳什均衡的存在与逼近

SIAM控制与优化杂志，第58卷，第3期，第1257-1288页，2020年1月。
我们考虑一类具有多维奇异控制的$ {N} $玩家随机游戏，其中每个玩家都面临带有亚模块成本的单调跟随器类型的最小化问题。我们称这些游戏为单调跟随游戏。在不一定是马尔可夫环境下，我们建立了纳什均衡的存在。此外，我们通过将\ mathbb {N} $中的每个$ n \限制为Lipschitz常数为$ n $的Lipschitz进程的玩家的可接纳策略，来引入一系列近似游戏。我们证明，对于\ mathbb {N} $中的每个$ n \，都存在一个近似博弈的Nash平衡，并且在Meyer-Zheng的意义上，纳什均衡的序列收敛于一个弱（分布）纳什。原始控制权博弈的均衡。作为副产品，这样的收敛还提供了两类游戏之间均衡值的近似结果。最后，我们展示了如何使用我们的结果来证明具有奇异控制的$ N $玩家随机微分游戏中开环Nash均衡的存在，并且我们提出了一种算法来确定单调跟随游戏的Nash平衡。