Splitting infinity: a de Finetti game with state-dependent profit rates and singular control for diffusions

TL;DR

This paper analyzes a resource extraction game with diffusive dynamics, characterizing non-trivial equilibria where players extract based on state-dependent thresholds, including novel control mechanisms like local time reflection and skew points.

Contribution

It introduces the first characterization of non-trivial equilibria with state-dependent thresholds and singular controls, including local time and skew points, in a resource extraction game.

Findings

Existence of non-trivial equilibria with threshold strategies

Characterization of equilibria involving local time reflection

Introduction of skew points with singular control increases

Abstract

We study a game of resource extraction of a common good under one-dimensional diffusive dynamics with player actions corresponding to singular stochastic control up to absorption at $0$, implying a trade-off between profitable resource extraction and sustainability. Unsurprisingly, immediate extraction of all available resources is an equilibrium. A main result is that we characterize and prove the existence of non-trivial equilibria that do not result in immediate absorption, but instead are attained with both players extracting resources according to a state-dependent rate of threshold type, corresponding to the presence of control only when the state process is in an interval $(b,\infty)$. The underlying assumption is, roughly, that the drift coefficient of the uncontrolled state process grows sufficiently fast in relation to the discount rate, implying that the value for the corresponding one-player problem is infinite. We also study a generalization of the game that allows a state-dependent profit rate integrated against the control processes. In this game we again characterize and prove the existence of non-trivial equilibria of threshold type. In particular, a main novelty is that we find equilibria where the state process is controlled with its own local time such that we have reflection points with associated initial jumps, as well as other points in the state space where the control processes increase in a singular manner (skew points).