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

Piotr Chlebicki, Kristoffer Lindensjö

TL;DR

This paper analyzes a two-player de Finetti-type stochastic game of resource extraction under one-dimensional diffusion dynamics with absorption at zero. It extends classical results by allowing highly general Markov controls that include jumps and local-time pushes (skew points), and it introduces a state-dependent profit rate through a function g. The authors establish non-trivial symmetric threshold equilibria where extraction occurs only above a barrier, provide verification and existence theorems under suitable assumptions, and generalize to the case of a discontinuous, state-dependent profit rate with skew points. They also present a case study featuring a bang-bang equilibrium with reflection at a barrier and an initial jump to that barrier, illustrating the richness of equilibria achievable with the generalized control framework. These results broaden the understanding of nonzero-sum stochastic control games and offer concrete structures for equilibria in diffusion-based resource-extraction models.

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).

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

TL;DR

This paper analyzes a two-player de Finetti-type stochastic game of resource extraction under one-dimensional diffusion dynamics with absorption at zero. It extends classical results by allowing highly general Markov controls that include jumps and local-time pushes (skew points), and it introduces a state-dependent profit rate through a function g. The authors establish non-trivial symmetric threshold equilibria where extraction occurs only above a barrier, provide verification and existence theorems under suitable assumptions, and generalize to the case of a discontinuous, state-dependent profit rate with skew points. They also present a case study featuring a bang-bang equilibrium with reflection at a barrier and an initial jump to that barrier, illustrating the richness of equilibria achievable with the generalized control framework. These results broaden the understanding of nonzero-sum stochastic control games and offer concrete structures for equilibria in diffusion-based resource-extraction models.

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 , 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 . 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).

Paper Structure

This paper contains 14 sections, 12 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and initial observations
  3. Admissible control strategies and global Markovian Nash equilibrium
  4. Trivial equilibria
  5. Assumptions and the one-player problem in our setting
  6. Equilibria with symmetric extraction rates above a threshold
  7. Generalizing to a state-dependent profit rate
  8. Equilibria with symmetric extraction rates and skew points above a threshold
  9. Proof of Theorem \ref{['Theorem_verification_g_NE_2']}
  10. Proof of Theorem \ref{['Theorem_existence_g_NE_1']}
  11. Examples
  12. A case study for equilibria with jumps and reflection
  13. Results for increasing fundamental solutions
  14. Proof of Theorem \ref{['Example-with-jumps']}

Key Result

Theorem 2.6

A pair of admissible control strategies $(D^1, D^2)\in \mathbb{L}\times\mathbb{L}$ satisfying $B^{i} = [0, \infty)$ and ${\cal J}^{i}(x)=x$ (which implies that $D^i_0= x$ and $\tau=0$ a.s., for any $X_{0-}=x$), is a global Markovian Nash equilibrium. The corresponding equilibrium values are given by

Figures (3)

  • Figure 1: Illustrations for Example \ref{['first-example']}, for for the parameters $r=0.08,\mu=0.25$ and $\sigma=2$. The value function $V_b$ is increasing in $x$ as well as in $b$. The extraction rate $\lambda_b^\ast$ is increasing in $x$ and decreasing in $b$.
  • Figure 2: Illustrations for Example \ref{['Example:g_jump']} (based on standard numerical methods, cf. Example \ref{['first-example']}) for $r=0.08$, $\mu=0.25$, $\sigma=2$, $\ell = 10$, $a_1 = a_2 = \frac{1}{2}$ (this corresponds to the $g$ in Remark \ref{['remark:motivating-example']}). The value function $V_b$ (given by \ref{['eq:newVb']}) is increasing in $x$ and in $b$, and there is a discontinuity in $b\mapsto V_b$ for $b=\ell$ (where the profit rate $g$ has its discontinuity). The extraction rate $\lambda_b^\ast$ is increasing in $x$ and decreasing in $b$ with a discontinuity at $x=b$ due to the indicator function (cf. \ref{['eq:lambda-inthmwithg']}), while the other discontinuities appear due to discontinuities in $x \mapsto V_b'(x)$ and $b \mapsto V_b(x)$. For $b < \ell$ the extraction equilibrium strategy includes a skew point (not illustrated here) with intensity $c_1=\frac{1}{3}$ at $x=\ell$ (calculated using \ref{['eq:def-c_j']}), while there is no skew point in case $b\geq\ell$.
  • Figure 3: Illustrations for Example \ref{['Example:g_complicated']}. The value function $V_b=G$ (cf. $b=0$), and the profit rate function $g$ are illustrated in the first panel. The second panel illustrates the skew points and their intensities as well as the extraction rate of the equilibrium strategy.

Theorems & Definitions (39)

  • Definition 2.1: Admissible control strategies
  • Remark 2.2: Interpretation of admissible control strategies and the corresponding controlled SDE \ref{['SDE']}
  • Definition 2.3: Expected rewards
  • Remark 2.4
  • Definition 2.5: Global Markovian Nash equilibrium
  • Theorem 2.6
  • proof
  • Remark 2.7: On the one-player problem in our setting
  • Remark 3.1
  • Theorem 3.2: Verification
  • ...and 29 more