# Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria

*Jodi Dianetti* and
*Giorgio Ferrari*

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Jodi Dianetti: Center for Mathematical Economics, Bielefeld University

Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University

No 605, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

**Abstract:**
We consider a class of N-player stochastic games of multi-dimensional 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 ∈ ℕ, the players' admissible strategies to the set of Lipschitz processes with Lipschitz constant bounded by n. We prove that, for each n ∈ 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.

**Keywords:** nonzero-sum games; singular control; submodular games; Meyer-Zheng topology; maximum principle; Nash equilibrium; stochastic differential games; monotone-follower problem. (search for similar items in EconPapers)

**Date:** 2019-01-09

**New Economics Papers:** this item is included in nep-gth

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https://pub.uni-bielefeld.de/download/2932994/2932998 First Version, 2019 (application/pdf)

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**Persistent link:** https://EconPapers.repec.org/RePEc:bie:wpaper:605

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