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

Dianetti J, Ferrari G (2019) Center for Mathematical Economics Working Papers; 605.
Bielefeld: Center for Mathematical Economics.

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Abstract / Bemerkung
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.
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Dianetti J, Ferrari G. Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria. Center for Mathematical Economics Working Papers. Vol 605. Bielefeld: Center for Mathematical Economics; 2019.
Dianetti, J., & Ferrari, G. (2019). Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria (Center for Mathematical Economics Working Papers, 605). Bielefeld: Center for Mathematical Economics.
Dianetti, J., and Ferrari, G. (2019). Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria. Center for Mathematical Economics Working Papers, 605, Bielefeld: Center for Mathematical Economics.
Dianetti, J., & Ferrari, G., 2019. Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria, Center for Mathematical Economics Working Papers, no.605, Bielefeld: Center for Mathematical Economics.
J. Dianetti and G. Ferrari, Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria, Center for Mathematical Economics Working Papers, vol. 605, Bielefeld: Center for Mathematical Economics, 2019.
Dianetti, J., Ferrari, G.: Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria. Center for Mathematical Economics Working Papers, 605. Center for Mathematical Economics, Bielefeld (2019).
Dianetti, Jodi, and Ferrari, Giorgio. Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria. Bielefeld: Center for Mathematical Economics, 2019. Center for Mathematical Economics Working Papers. 605.
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