Cooperative games defined by multi-objective optimization in competition for subsurface resources

TL;DR

The paper develops a novel framework that casts competition for subsurface resources as cooperative games in partition-function form with externalities, where each coalition's value is a Pareto-optimal solution to a constrained multi-objective optimization problem. A central theoretical result shows that the Pareto set for a coarser coalition structure contains the Pareto sets for all finer refinements, enabling hierarchical, cost-effective computation of all coalition-structure Pareto fronts from the singleton structure. The authors demonstrate the approach on groundwater management and large-scale CO$_2$ storage (Bjarmeland, Barents Sea), using NSGA-II and CSO to solve MOOs and validate that hierarchical MOO can substantially reduce the number of physical-model evaluations while capturing a wide range of externalities (negative, zero, mixed). The framework supports a-posteriori PF selection and various decision criteria, offering a flexible, site-specific tool for orchestrating collaboration among subsurface-activity actors and informing policy or contract design. Overall, this work provides both theoretical insight into the structure of physics-informed cooperative games and practical numerical strategies to enable efficient, decision-relevant exploration of potential collaborations.

Abstract

We propose a novel decision making framework for forming potential collaboration among otherwise competing agents in subsurface systems. The agents can be, e.g., groundwater, CO$_2$, or hydrogen injectors and extractors with conflicting goals on a geophysically connected system. The operations of a given agent affect the other agents by induced pressure buildup that may jeopardize system integrity. In this work, such a situation is modeled as a cooperative game where the set of agents is partitioned into disjoint coalitions that define the collaborations. The games are in partition function form with externalities, i.e., the value of a coalition depends on both the coalition itself and on the actions of external agents. We investigate the class of cooperative games where the coalition values are the total injection volumes as given by Pareto optimal solutions to multi-objective optimization problems subject to arbitrary physical constraints. For this class of games, we prove that the Pareto set of any coalition structure is a subset of any other coalition structure obtained by splitting coalitions of the first coalition structure. Furthermore, the hierarchical structure of the Pareto sets is used to reduce the computational cost in an algorithm to hierarchically compute the entire Pareto fronts of all possible coalition structures. We demonstrate the framework on a pumping wells groundwater example, and nonlinear and realistic CO$_2$ injection cases, displaying a wide range of possible outcomes. Numerical cost reduction is demonstrated for the proposed algorithm with hierarchically computed Pareto fronts compared to independently solving the multi-objective optimization problems.

Figures (14)

• Figure 1: Coalition structure graphs for (a) three agents; and (b) four agents, respectively. Following the edges colored in red from a target coalition structure $\{ \{a_1, a_3\}, \{a_2, a_4\} \}$ shows the coalition structures whose corresponding Pareto sets are supersets of the target Pareto set, i.e., $\{\{a_1, a_3\}, \{a_2\}, \{a_4\}\}$, $\{\{a_1\}, \{a_2, a_4\}, \{a_3\} \}$, and $\{\{a_1\}, \{a_2\}, \{a_3\}, \{a_4\}\}$.
• Figure 2: Overview of full workflow with class of cooperative games with value functions defined by the solutions to multi-objective optimization problems with physical constraints.
• Figure 3: Pareto fronts for the groundwater problem with three agents. The social welfare maximizing solution is indicated by a red marker. For ease of notation, we only indicate the the coalition $C$ of the value function, but the correct full notation is the embedded coalition $(C,CS)$.
• Figure 4: Externalities for the groundwater problem with three agents. (a) Negative externalities; (b) mixed externalities; (c) zero externalities; (d) all externalities.
• Figure 5: Total injection volumes for the weighted utopia point selection criterion, Eq. \ref{['eq:utopia_norm']}, with coalition weights \ref{['eq:weighted_agents']} given by (a) $\tilde{\beta} = [0.97, 0.01, 0.01, 0.01]$, (b) $\tilde{\beta} =[0.45, 0.45, 0.05, 0.05]$, and (c) $\tilde{\beta} = [0.25, 0.25, 0.25, 0.25]$.

Theorems & Definitions (10)

• Definition 1: Coalition structure
• Definition 2: Characteristic function game
• Definition 3: Partition function game
• Lemma 1
• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Remark 4