Sustainable Exploitation Equilibrium (SEE)

• Sustainable Exploitation Equilibrium (SEE) is a formal solution concept that balances maximal resource extraction with strict non-collapse and long-run viability constraints.
• It employs optimal control, ergodic payoff maximization, and equilibrium strategies to maintain resource stocks within defined sustainability sets.
• In dynamic games and multi-agent systems, SEE refines traditional equilibria by enforcing renegotiation-proof and Pareto-efficient strategies that prevent coordinated deviation.

A Sustainable Exploitation Equilibrium (SEE) is a formal solution concept emerging across resource economics, stochastic control, dynamic games, and applied systems analysis, defined by the requirement that resource extraction or exploitation is maximized subject to strict non-collapse or long-run viability constraints. It is typically characterized by optimal control or equilibrium strategies that maximize long-term or ergodic payoffs, ensure state variables (e.g., resource stocks, ecosystem health, exploited agents) remain within a predefined sustainability set, and—in multi-agent or game-theoretic formulations—additionally enforce stability against renegotiation or collective deviation. The SEE concept unifies prior approaches in optimal harvesting, viability theory, and sustainable game theory under a mathematically rigorous framework suitable for applications in natural resource management, environmental stewardship, economic development, and even multi-agent artificial intelligence.

1. Mathematical Formulations of SEE in Optimal Control and Stochastic Systems

In deterministic and stochastic resource management, SEE arises as the optimal long-run average or ergodic payoff attained under admissible harvesting strategies that prevent resource collapse. For singular stochastic control problems, as in (Liang et al., 2020), the resource dynamics are modeled as a controlled diffusion process: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t - d\zeta_t,\quad X_{0-}=x>0,$$ where $X_t$ is the resource stock and $\zeta_t$ is the cumulative extraction (monotone, nondecreasing, càdlàg, adapted, ensuring $X_t>0$ for all $t$).

The SEE payoff is defined as the value of the ergodic control problem: $$J_x^E(\zeta) = \limsup_{T\to\infty}\frac{1}{T}\left[\int_0^T h(X_t^\zeta)dt + \int_0^T k(X_t^\zeta)\circ d\zeta_t\right],$$ where $h(x)$ is ecosystem utility and $k(x)$ is marginal harvest profit.

The associated stationary Hamilton–Jacobi–Bellman quasi-variational inequality (HJB–QVI) with gradient constraint is: $$\max\left\{ \mathcal{L}w(x) + h(x) - \lambda,\; k(x) - w'(x) \right\} = 0, \quad x>0,$$ where $\lambda$ is the ergodic payoff and $\mathcal{L}$ is the Ito differential operator. The optimal exploitation strategy reflects resource stock at a unique threshold $b^*$, found via smooth-fit (free-boundary) conditions: $$\lambda^* = \frac{1}{m(0,b^*)}\int_0^{b^*}[K(s)+h(s)]m(ds), \quad K(b^*)+h(b^*)-\lambda^*=0,$$ yielding a unique solution $(b^*,\lambda^*)$ under weak regularity assumptions. The resulting SEE harvest policy maintains $X_t\in[0,b^*]$ and delivers sustainable payoff $\lambda^*$ (Liang et al., 2020).

2. Dynamic Game-Theoretic Refinements and Renegotiation-Proof SEE

Expanding into dynamic games, SEE provides a strict refinement of Markov Perfect Equilibrium (MPE) for systems with exploiter–exploitee structure (Kirk, 8 Dec 2025). In these models, the sustainability set $\mathcal{S}\subset S$ restricts feasible state-action trajectories: $$f(s,x,e)\in\mathcal{S}\quad\forall\,s\in\mathcal{S},\,x\in X(s),\,e\in E(s)$$ where $x$ and $e$ are actions of the exploiter and exploitee, and $f$ is the deterministic transition.

A profile $\sigma^*$ of stationary strategies is an SEE if:

• (i) Viability: State never leaves $\mathcal{S}$.
• (ii) Renegotiation-proofness: $\sigma^*$ is not Pareto-dominated (in value functions) by any other viable MPE.
• (iii) Exploiter-optimal selection: Among all renegotiation-proof, viable profiles, $\sigma^*$ maximizes the exploiter's continuation value.

The existence of such profiles follows from fixed-point and Pareto-front arguments under compactness and continuity conditions. The SEE is thus a refinement satisfying both viability and immunity to mutually beneficial renegotiation, ensuring strictly sustainable exploitation in dynamic strategic settings. This structure is illustrated in hegemon–client models and extends to general dynamic resource games (Kirk, 8 Dec 2025).

3. SEEs in Common-Pool Resource Models and Multi-Agent Systems

SEE is formally specified for common-pool resources in both analytic and simulation environments. In discrete-time multi-agent settings, such as the GovSim environment (Piatti et al., 25 Apr 2024), the SEE is defined as a steady-state $(S^*,e^*)$ such that:

• Resource dynamics: $$S^* = S^* + R(S^*) - I\,e^* \implies R(S^*) = I\,e^*$$
• Extraction uniformity: $e_i^t = e^*$ for all agents and all $t$
• Non-collapse: $S^* > C$ for collapse threshold $C$

For logistic regeneration, $R(S)=rS(1-S/K)$, the equilibrium conditions reduce to

$rS^*(1-S^*/K) = I\,e^*, \quad S^*\in(C,K)$

with stability enforced via $|1 + R'(S^*)| < 1$. The dynamic protocol requires memory, transparent extraction reporting, negotiation, and application of universalization reasoning—prompting agents to forecast the effect of uniform over-extraction and coordinate on $e^* = R(S^*)/I$. Empirical results demonstrate only advanced agents with explicit universalized reasoning robustly converge to SEE, and perturbations (e.g. introduction of greedy agents) are mitigated by negotiated re-entry to SEEs (Piatti et al., 25 Apr 2024).

4. Analytical Characterizations of SEE in Bioeconomic and Ecosystem Models

Multiple natural resource and ecological models implement SEE as a "bioeconomic" or dynamic equilibrium balancing exploitation and sustainability:

• In harvest models with tipping points (To, 2 Apr 2025), SEE corresponds to the unique high-fecundity steady state $x^*\geq x_p$ solving $f'(x^*) = (1/\beta) - 1$, with $h^* = f(x^*) - x^*$. Below critical Skiba thresholds, the system converges to a less productive, low-fecundity steady state. Hysteresis introduces meta-stability, with irreversible transitions when tipping points are crossed.
• In size-structured fisheries models (Filar et al., 2020), SEE is defined explicitly by the square-root law: optimal escapement rate $\gamma^* = 1/\sqrt{\rho}$ (with proliferation rate $\rho$) achieves maximal sustainable yield, and the corresponding equilibrium biomass $\bar x^* = (\sqrt{\rho}-1)/(\rho-1)K$ is operationalized as the sustainable equilibrium abundance.
• In predator–prey or multi-trophic systems (Panayotova et al., 2022), SEE ("bionomic equilibrium") is a joint steady state $(x^*,y^*,E_1^*,E_2^*)$ with positive abundances and zero long-run economic rent, derived from the intersection of stationary population and economic rent conditions.
• In orbital resource management (Gkolias et al., 2019), SEE demarcates coexistence of long-lived "graveyard" orbits and natural re-entry highways, ensuring the non-accumulation of debris in protected geosynchronous belts. Quantitative orbital element limits and resonance exploitation define the SEE domain.

5. Multicriteria SEEs in Socio-Ecological and System-Dynamics Models

In applied contexts with multiple interacting objectives (ecological, economic, social), the SEE is defined as a Pareto-efficient equilibrium of a constrained nonlinear dynamic system. In overtourism management (Lyu et al., 18 Nov 2025), the SEE is a tuple $(x^*,g^*,e^*,s^*)$ (tourist arrivals, cumulative revenue, environment quality, social satisfaction) lying on the non-dominated frontier of a multi-objective system: $$(x_{t+1},g_{t+1},e_{t+1},s_{t+1})=F(x_t,g_t,e_t,s_t;d)$$ where $d$ are policy levers (tax, carbon fee, capacity, investment rates). The SEE is characterized by equilibrium, feasibility (e.g., $e^*,s^*\in[0,1]$), local stability (Jacobian spectral conditions), and optimality with respect to NSGA-II derived Pareto fronts. Sensitivity analysis identifies key policy parameters (e.g., carbon fees, capacity limits) dominating the SEE outcomes.

6. Broader Significance and Key Theoretical Insights

The SEE concept represents a universal refinement for the sustainable management of dynamical systems involving extractive pressures and critical collapse thresholds. Across all mathematical traditions in which it appears:

• SEE imposes hard constraints on viability—excluding exploitation trajectories leading outside the sustainability set
• Equilibrium (stationarity or ergodicity) is coupled with local or global stability conditions, often involving spectral criteria or eigenvalue bounds on system Jacobians
• In game-theoretic or multi-agent settings, SEE refines Nash/MPE with renegotiation-proofness, incentive-alignment, and exploitative optimality (or by non-dominated trade-offs in multicriteria environments)
• Quantitative thresholds (e.g., resource stocks, escapement rates, economic saturation points, orbital element limits) are determined endogenously by system dynamics, subject to exogenous feasibility conditions

Empirical and simulation studies using SEE—spanning fisheries, renewable resources, orbital debris, AI cooperative agents, and tourism economies—demonstrate that enforcing SEE policies can provide robust guardrails against collapse, clarify inherent trade-offs, and formalize stability against both perturbations and coordinated deviation.

7. Table: Representative SEE Frameworks and Core Criteria

Context / Model Type	SEE Characterization	Canonical Reference
Stochastic control (diffusions)	Threshold reflection at $b^*$, ergodic payoff $\lambda^*$	(Liang et al., 2020)
Dynamic games (MPE refinement)	Viable, renegotiation-proof, exploiter-optimal profiles in $\mathcal{S}$	(Kirk, 8 Dec 2025)
Multi-agent common resource	Stationary $(S^*,e^*)$ with $R(S^*)=I\,e^*$, $S^*>C$, stability	(Piatti et al., 25 Apr 2024)
Harvest with tipping/hysteresis	High-fecundity steady-state, $f'(x^*)=(1/\beta)-1$, $x^*>x_p$	(To, 2 Apr 2025)
Structured fisheries (square root)	Escapement $\gamma^*=1/\sqrt{\rho}$, equilibrium abundance $\bar x^*$	(Filar et al., 2020)
Predator–prey bioeconomics	Equilibrium with bioeconomic rents $=0$, positive stock/effort	(Panayotova et al., 2022)
Multicriteria system dynamics	Pareto-efficient equilibrium, all constraints active, local stability	(Lyu et al., 18 Nov 2025)

Each instantiation of SEE is rigorously defined (model-specific equilibrium, viability, uniqueness, and stability criteria) and concretely implementable in either analytic, algorithmic, or simulation-based resource management.