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In this paper, we deal with the solutions of systems of PDEs with bilateralinter-connected obstacles of min-max and max-min types. These systems arisenaturally in stochastic switching zero-sum game problems. We show that when theswitching costs of one side are regular, the solutions of the min-max andmax-min systems coincide. Then, this common viscosity solution is related to amulti-dimensional doubly reflected BSDE with bilateral interconnectedobstacles. Finally, its relationship with the the values of a zero-sumswitching game is studied.

On the Equality of Solutions of Max-Min and Min-Max Systems of Variational Inequalities with Interconnected Bilateral Obstacles