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Sustainable Exploitation Equilibria for Dynamic Games with Irreversible Failure

Nicholas H. Kirk

Abstract

We study dynamic relationships in which one party extracts current surplus in ways that degrade the future state, while the counterparty cannot exit but adjusts effort in response. Standard stationary Markov equilibria may sustain collapse paths in which short-run extraction dominates strictly positive continuation gains. We introduce the Sustainable Exploitation Equilibrium (SEE), a refinement for dynamic games with irreversible failure modeled as an absorbing boundary that eliminates continuation value. When a survival-preserving action exists and failure destroys future surplus, equilibria assigning positive probability to collapse are sequentially irrational. Equilibrium analysis can therefore be restricted, without loss, to continuation-preserving stationary Markov equilibria. Within this restricted domain, viable renegotiation-proofness becomes structural: because failure truncates future surplus, any jointly improving survival-preserving deviation is credible prior to collapse. SEE selects the viable, renegotiation-stable equilibrium that maximizes the exploiter's value. Existence is established under standard conditions, and the refinement is illustrated in a hegemon-client setting.

Sustainable Exploitation Equilibria for Dynamic Games with Irreversible Failure

Abstract

We study dynamic relationships in which one party extracts current surplus in ways that degrade the future state, while the counterparty cannot exit but adjusts effort in response. Standard stationary Markov equilibria may sustain collapse paths in which short-run extraction dominates strictly positive continuation gains. We introduce the Sustainable Exploitation Equilibrium (SEE), a refinement for dynamic games with irreversible failure modeled as an absorbing boundary that eliminates continuation value. When a survival-preserving action exists and failure destroys future surplus, equilibria assigning positive probability to collapse are sequentially irrational. Equilibrium analysis can therefore be restricted, without loss, to continuation-preserving stationary Markov equilibria. Within this restricted domain, viable renegotiation-proofness becomes structural: because failure truncates future surplus, any jointly improving survival-preserving deviation is credible prior to collapse. SEE selects the viable, renegotiation-stable equilibrium that maximizes the exploiter's value. Existence is established under standard conditions, and the refinement is illustrated in a hegemon-client setting.

Paper Structure

This paper contains 31 sections, 1 theorem, 12 equations.

Table of Contents

  1. Introduction
  2. Why the refinement is necessary under irreversible failure.
  3. Related Literature
  4. Equilibrium refinements.
  5. Viability and dynamic control.
  6. Political economy and relational contracts.
  7. Comparison.
  8. Model and Definition
  9. Environment and timing.
  10. Equilibrium notion (Markov--Stackelberg Equilibrium).
  11. Rational interpretation of viability.
  12. Multiplicity and uniqueness.
  13. Relation to Nash and Markov Perfect Equilibrium
  14. Nash equilibrium (NE).
  15. Subgame perfect equilibrium (SPE).
  16. ...and 16 more sections

Key Result

Theorem 1

Suppose the state space $S$ and action sets $X(s), E(s)$ are compact, payoffs are bounded and continuous, and the transition kernel $Q$ satisfies the Feller property. Interpret $S \setminus V$ as an absorbing failure region that eliminates continuation value. Then there exists $M^\star < \infty$ suc is viable. Moreover, among viable stationary Markov Perfect Equilibria, a nonempty compact subset o

Theorems & Definitions (5)

  • Definition 1: Exploiter
  • Definition 2: Exploitee (no exit, effort choice)
  • Definition 3: Sustainable Exploitation Equilibrium (SEE)
  • Theorem 1: Existence of SEE under rational viability
  • proof