Co-opetition Equilibrium

• Co-opetition Equilibrium (CoE) is a formal framework for modeling systems where agents simultaneously engage in cooperation and competition through interdependent utility functions.
• Its applications span continuous-action games, adversarial team setups, coalition-resource models, and mixed cooperative-competitive markets.
• The framework quantifies interdependence and synergy, providing actionable insights into equilibrium computation, stability, and comparative statics in complex systems.

Co-opetition Equilibrium (CoE) is a formal solution concept for analyzing systems where agents simultaneously engage in cooperative and competitive interactions—in short, situations of "coopetition." CoE frameworks interpolate between full cooperation and pure competition depending on structural dependencies, complementarity, and the heterogeneity of agent objectives. The concept has been rigorously developed in continuous-action games with interdependent utilities, in adversarial team games with heterogeneous team members, within multi-agent coalition-formation models, and in markets comprising both cooperative and non-cooperative populations (Pant et al., 21 Oct 2025, Zhang, 22 Dec 2025, Fujii, 2022, Singhal, 2023). CoE provides an equilibrium foundation for modeling and predicting agent behavior in socio-technical and strategic environments characterized by nontrivial blends of value creation and value capture.

1. Core Formalizations of Co-opetition Equilibrium

The mathematical formalization of CoE varies by domain, but a common thread is augmenting agents' classical objective functions to internalize interdependence, synergistic value creation, and incentive compatibility within coalitions or teams.

Continuous-action interdependent games

Let $N$ be a set of actors, each choosing $a_i \in \mathbb R_+$. Actor $i$’s utility function is

$$U_i(\mathbf a) = \pi_i(\mathbf a) + \sum_{j \neq i} D_{ij} \, \pi_j(\mathbf a),$$

where $\pi_i(\mathbf a)$ is the private payoff and $D_{ij} \in [0,1]$ encodes quantitative dependence of $i$ on $j$. A Co-opetition Equilibrium is any fixed point $\mathbf a^* = (a_1^*, ..., a_N^*)$ such that

$$a_i^* \in \arg\max_{a_i \ge 0}\; U_i(a_i, \mathbf a_{-i}^*),$$

for every $i$ (Pant et al., 21 Oct 2025).

Adversarial team games with heterogeneity

Given $N$ players, the team $T = N \setminus \{n\}$ faces adversary $n$. Let $X_T$ and $X_n$ denote team and adversary (possibly correlated) mixed strategies, and $u_i$ the (potentially heterogeneous) utility for each team member. A CoE is a pair $(x_T, x_n)$ such that:

• The adversary plays a best response: $u_n(x_T, x_n) \ge u_n(x_T, y_n)$ for all $y_n$.
• The team plan $x_T$ is a correlated equilibrium relative to $x_n$: all incentive-compatibility constraints hold for each $i \in T$ wrt their own $u_i$ (Zhang, 22 Dec 2025).

Coalition-resource games and population models

In coalition models, a CoE is a partition and allocation $(P, \phi)$ such that

• Feasibility: each coalition gets its joint equilibrium value from the resource-sharing game.
• Unilateral or restricted-blocking stability: no player or group can improve by splitting or merging, given the competitive constraints (Singhal, 2023).

In mean-field markets, CoE refers to the unique equilibrium that emerges when both cooperative (centrally planned) and non-cooperative (price-taking) populations interact, subject to market-clearing and cost minimization (Fujii, 2022).

2. Structural Interdependence and Quantitative Modeling

Quantifying interdependence is central to CoE. In structural modeling (i* framework), dependencies are mapped as follows: $$D_{ij} = \frac{\sum_{d \in \mathcal D_i} w_d\, \text{Dep}(i, j, d)\, \text{crit}(i, j, d)}{\sum_{d \in \mathcal D_i} w_d},$$ where $w_d$ is the importance of dependum $d$, $\text{Dep}(i, j, d)$ signals a dependence link, and $\text{crit}(i, j, d)$ the degree of criticality or substitutability (Pant et al., 21 Oct 2025).

This parameterization grounds interdependence in the system’s graphical structure and enables integration of qualitative requirements models with quantitative game-theoretic predictions.

3. Value Creation, Complementarity, and Appropriation

Value creation in CoE frameworks synthesizes additivity and synergy: $$V(\mathbf a \mid \gamma) = \sum_{i=1}^N f_i(a_i) + \gamma\,g(a_1, ..., a_N),$$ with $f_i$ representing standalone returns (e.g., $a_i^\beta$, $\beta \in (0,1)$, or $\theta \log(1+ a_i)$) and $g$ quantifying symmetric synergy (e.g., geometric mean) (Pant et al., 21 Oct 2025). The complementarity parameter $\gamma$ controls superadditive gains from interaction; higher $\gamma$ yields more pronounced co-creation of value.

Individual payoff appropriation incorporates bargaining shares $\alpha_i$: $$\pi_i(\mathbf a) = e_i - a_i + f_i(a_i) + \alpha_i [V(\mathbf a) - \sum_j f_j(a_j)],$$ with $\alpha_i$ proportional to a bargaining power parameter $\beta_i$: $\alpha_i = \frac{\beta_i}{\sum_j \beta_j}$

This structure captures both the creation (via $V$) and the capture (via $\alpha_i$ and $D_{ij}$) of value in coopetitive contexts.

4. Existence, Uniqueness, and Equilibrium Properties

Existence of CoE follows from the application of standard fixed-point arguments (Kakutani). For interdependence-augmented Nash games, the best response correspondence is upper hemicontinuous and nonempty under compactness and continuity. Uniqueness requires additional conditions, typically strict concavity of $U_i$ and contraction of the best-response mapping (Pant et al., 21 Oct 2025). For large-population mean-field models, uniqueness holds if the share of the "cooperative" group is below a critical threshold (Fujii, 2022).

In adversarial team games, the existence of CoE is guaranteed by reduction to Nash equilibrium; every Nash profile yields a CoE by marginalization. CoE is not unique in general—multiplicity arises as in classic Nash settings. Special structure (zero-sum, consistent team weights) enables stronger properties, such as exchangeability and tractability (Zhang, 22 Dec 2025).

For coalition models, stability concepts (U-core, RB-IA) ensure that specific partitions are robust to various deviation modes (Singhal, 2023).

5. Algorithms, Complexity, and Comparative Statics

The computational complexity of CoE computation depends on the model structure:

• General interdependent games: Computing CoE is as hard as finding Nash equilibria; PPAD-complete.
• Team-maximizing CoE (TMCoE): Maximizing aggregate team payoff among CoEs is NP-hard even for simple adversarial team games; solvable as a nonconvex program (Zhang, 22 Dec 2025).
• Zero-sum consistent utility case: TMCoE is equivalent to a two-player zero-sum equilibrium and tractable via linear programming.
• In coalition games, enumeration and deviation-checking determine the stability of candidate partitions; explicit formulas are available for symmetric and asymmetric agent parameters (Singhal, 2023).
• In mean-field models, coupled forward-backward SDE systems or corresponding PDEs characterize the limiting CoE; convergence from the finite-agent equilibrium is established under regularity conditions (Fujii, 2022).

Comparative statics elucidate how increases in interdependence $D_{ij}$, complementarity $\gamma$, or cooperative group size $\theta$ shift equilibrium towards greater cooperation, higher joint surplus, or potentially unstable market configurations.

6. Applications and Empirical Validation

CoE frameworks have been empirically validated and applied across domains:

• In the Samsung–Sony S-LCD joint venture case, empirical calibration of interdependence coefficients and value-creation functions demonstrated CoE’s predictive accuracy for real coopetitive investments. The logarithmic standalone utility specification provided superior empirical fit (validation score $45/60$ over $30/60$ for power functions), confirming robustness across functional forms (Pant et al., 21 Oct 2025).
• In resource allocation scenarios, CoE characterizes stable coalition structures and predicts value division in resource-constrained environments, reconciling cooperative scale economies with competitive resource contention (Singhal, 2023).
• In financial markets, equilibrium price formation is governed by the joint (cooperative group) and individual (competitive agent) trading dynamics, with CoE marking the unique market-clearing outcome when agents span both populations. Instability emerges as the cooperative share exceeds model-specific bounds (Fujii, 2022).
• In adversarial team games, CoE resolves the instability and inefficiency that arise when agent heterogeneity is ignored, and supports coordination among diverse coalitional partners facing opposition (Zhang, 22 Dec 2025).

7. Limitations and Open Research Directions

Several open directions and limitations persist:

• Exact computation of CoE is computationally intractable in general; practical approaches must exploit problem structure or employ approximation and heuristic methods.
• CoE existence breaks down in certain multi-adversary or multi-team contexts unless payoff structures exhibit strong consistency.
• Most CoE theory remains confined to normal-form and static games; the extension to extensive-form or dynamic interactions is open.
• The characterization of tractable payoff classes and the efficient learning of approximate CoE in large-scale systems represent active research frontiers.
• Empirical analysis suggests that model misspecification in utility functions or interdependence structure can materially affect the stability and predictive power of the CoE framework.

A plausible implication is that future work must balance model tractability with adequate representation of agent heterogeneity and structural interdependence to apply CoE methodology effectively in real-world multi-agent systems (Zhang, 22 Dec 2025, Pant et al., 21 Oct 2025).

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Key References:

• "Computational Foundations for Strategic Coopetition: Formalizing Interdependence and Complementarity" (Pant et al., 21 Oct 2025)
• "Considering the Difference in Utility Functions of Team Players in Adversarial Team Games" (Zhang, 22 Dec 2025)
• "Equilibrium pricing of securities in the co-presence of cooperative and non-cooperative populations" (Fujii, 2022)
• "Navigating Resource Conflicts: Co-opetition and Fairness" (Singhal, 2023)