Explicit solution to an optimal two-player switching game in infinite horizon

TL;DR

This work studies an infinite-horizon two-player switching game where the state $X$ follows a one-dimensional Itô diffusion and players alternate regimes at stopping times to maximize/minimize a discounted payoff. The authors employ viscosity-solution techniques to derive a system of quasi-variational inequalities and characterize switching regions via threshold values, turning the problem into a finite-threshold control problem. They provide an explicit solution in the symmetric-profit case with differing diffusion operators, using a base solution $rac{1}{r-b_{ij}gamma+rac12msa_{ij}^2gamma(1-gamma)}}x^{gamma}$ and a homogeneous component $A_{ij}x^{m^+_{ij}}+B_{ij}x^{m^-_{ij}}$, and they delineate switching-region structures under several cost scenarios (B1–B4) with smooth-fit across boundaries. A numerical procedure is proposed to determine threshold values when only qualitative region structure is known, supported by numerical simulations and graphical illustrations. Overall, the paper delivers an explicit, threshold-based value function solution for a two-player switching game, reducing reliance on iterative numerical methods and offering practical insight for applications in areas such as finance and energy. The approach lays groundwork for extending to non-identical profit functions and more complex state dynamics.

Abstract

In this paper we use viscosity approach to provide an explicit solution to the problem of a two - player switching game. We characterize the switching regions which reduce the switching problem into one of finding a finite number of threshold values in state process that would trigger switchings and then derive an explicit solution to this problem. The state process is a one dimensional Itô diffusion process and switching costs are allowed to be non-positive. We also suggest a numerical procedure to compute the value function in case we know the qualitative structure of switching regions and we illustrate our results by numerical simulations.

Figures (5)

• Figure 1: Graphic Illustration of Strategies in Theorem 4.3-ii
• Figure 2: Theorem 4.3 B2: $I_0=(1,1)$
• Figure 3: Theorem 4.3 B2: $I_0=(1,2)$
• Figure 4: Theorem 4.3 B2: $I_0=(2,1)$
• Figure 5: Theorem 4.3 B2: $I_0=(2,2)$

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Proof 2.1
• Lemma 2.2
• Proof 2.2
• Lemma 2.3
• Proof 2.3
• Lemma 3.1
• Proof 3.1