Parameter Reference
Validated Parameters for Coopetition-Gym Environments
This document provides a comprehensive reference for all parameters used in Coopetition-Gym, including validated values, acceptable ranges, and calibration guidance.
Quick Reference Card
For practitioners who need parameter values quickly:
Recommended Default Values
# Value Function Parameters (TR-1)
THETA = 20.0 # Logarithmic scale
BETA = 0.75 # Power exponent
GAMMA = 0.65 # Complementarity
# Trust Dynamics Parameters (TR-2)
LAMBDA_PLUS = 0.10 # Trust building rate
LAMBDA_MINUS = 0.30 # Trust erosion rate
MU_R = 0.60 # Reputation damage severity
DELTA_R = 0.03 # Reputation decay rate
XI = 0.50 # Interdependence amplification
KAPPA = 1.0 # Signal sensitivity
# Initial Conditions
T_INIT = 0.50 # Initial trust level
R_INIT = 0.00 # Initial reputation damage
Pillar 1: Interdependence & Complementarity Parameters
Value Function Parameters
Logarithmic Specification (Recommended)
| Parameter | Symbol | Validated Value | Range | Source |
|---|---|---|---|---|
| Scale Factor | θ | 20.0 | [15, 30] | TR-1 §7.3, S-LCD validation |
| Complementarity | γ | 0.65 | [0.50, 0.80] | TR-1 §7.2, multi-criteria |
Usage:
\[f_i(a_i) = \theta \cdot \ln(1 + a_i)\] \[V(\mathbf{a} \mid \gamma) = \sum_{i=1}^{N} f_i(a_i) + \gamma \cdot \left(\prod_{i=1}^{N} a_i\right)^{1/N}\]Validation Performance: 58/60 (96.7%) on Samsung-Sony S-LCD case study
When to Use: Manufacturing joint ventures, technology partnerships, scenarios where initial capabilities are highly valuable but incremental improvements have declining impact.
Power Specification (Alternative)
| Parameter | Symbol | Validated Value | Range | Source |
|---|---|---|---|---|
| Exponent | β | 0.75 | [0.65, 0.85] | TR-1 §7.1, 22,000+ trials |
| Complementarity | γ | 0.65 | [0.50, 0.80] | TR-1 §7.2 |
Usage:
\[f_i(a_i) = a_i^{\beta}\] \[V(\mathbf{a} \mid \gamma) = \sum_{i=1}^{N} f_i(a_i) + \gamma \cdot \left(\prod_{i=1}^{N} a_i\right)^{1/N}\]Validation Performance: 46/60 (76.7%) on Samsung-Sony S-LCD case study
When to Use: General scenarios, platform ecosystems, when cooperation magnitudes may be larger.
Comparison
| Criterion | Logarithmic (θ=20) | Power (β=0.75) | Winner |
|---|---|---|---|
| S-LCD Validation | 58/60 | 46/60 | Logarithmic |
| Cooperation Prediction | 41% increase | 166% increase | Logarithmic (realistic) |
| Bounded Returns | Yes | No | Logarithmic |
| Mathematical Tractability | Moderate | High | Power |
Interdependence Parameters
| Parameter | Symbol | Typical Values | Range | Description |
|---|---|---|---|---|
| Dependency Weight | $w_d$ | Context-specific | [0, 1] | Goal importance (normalized) |
| Criticality Factor | $\text{crit}$ | Calculated | [0, 1] | 1/n for n alternatives |
| Bargaining Share | $\alpha_i$ | 0.50 (symmetric) | [0, 1] | Must sum to 1.0 |
Interdependence Matrix Guidance:
| Relationship Type | $D_{ij}$ Range | Example |
|---|---|---|
| No dependency | 0.00 | Competitors in separate markets |
| Weak dependency | 0.10 - 0.30 | Multiple alternative suppliers |
| Moderate dependency | 0.30 - 0.60 | Preferred but substitutable partner |
| Strong dependency | 0.60 - 0.85 | Critical supplier, few alternatives |
| Complete dependency | 0.85 - 1.00 | Sole provider of essential resource |
Pillar 2: Trust Dynamics Parameters
Trust Evolution Parameters
| Parameter | Symbol | Validated Value | Range | Source |
|---|---|---|---|---|
| Trust Building Rate | $\lambda^+$ | 0.10 | [0.05, 0.15] | TR-2 §7.2 |
| Trust Erosion Rate | $\lambda^-$ | 0.30 | [0.20, 0.45] | TR-2 §7.2 |
| Negativity Ratio | $\lambda^-/\lambda^+$ | 3.0 | [2.5, 4.0] | Behavioral economics |
The 3:1 Ratio: Empirically grounded in behavioral economics research showing trust erodes approximately 3× faster than it builds. This captures:
- Negativity bias in human judgment
- Asymmetric impact of violations vs. cooperation
- Evolutionary caution toward potential threats
Calibration Guidance:
| Context | $\lambda^+$ | $\lambda^-$ | Ratio | Rationale |
|---|---|---|---|---|
| High-trust culture | 0.12 | 0.30 | 2.5 | Faster trust building |
| Standard business | 0.10 | 0.30 | 3.0 | Default validated values |
| Low-trust environment | 0.08 | 0.35 | 4.4 | Slower building, faster erosion |
| Post-crisis recovery | 0.06 | 0.25 | 4.2 | Difficult trust rebuilding |
Reputation Parameters
| Parameter | Symbol | Validated Value | Range | Source |
|---|---|---|---|---|
| Damage Severity | $\mu_R$ | 0.60 | [0.45, 0.75] | TR-2 §7.3 |
| Decay Rate | $\delta_R$ | 0.03 | [0.01, 0.05] | TR-2 §7.3 |
Interpretation:
- $\mu_R = 0.60$: A full violation ($s = -1$) causes 60% of available reputation space to be damaged
- $\delta_R = 0.03$: Approximately 33 periods of no violations to decay reputation by 63%
Hysteresis Effect:
\[\Large \Theta = 1 - R \quad \text{(Trust Ceiling)}\]| Stage | $T$ | $R$ | $\Theta$ | Status |
|---|---|---|---|---|
| Initial | 0.50 | 0.00 | 1.00 | Full recovery possible |
| After violation | 0.35 | 0.40 | 0.60 | Ceiling at 60% |
| After recovery | 0.58 | 0.40 | 0.60 | Cannot exceed ceiling |
Amplification Parameters
| Parameter | Symbol | Validated Value | Range | Source |
|---|---|---|---|---|
| Interdep. Amplification | $\xi$ | 0.50 | [0.30, 0.70] | TR-2 §7.4 |
| Signal Sensitivity | $\kappa$ | 1.0 | [0.5, 2.0] | TR-2 §6.1 |
Interdependence Amplification Effect:
\[\Large \text{Erosion} = \lambda^- \cdot |s| \cdot T \cdot (1 + \xi \cdot D_{ij})\]| Dependency Level | $D_{ij}$ | Erosion Factor |
|---|---|---|
| Low dependency | 0.2 | 1.10 |
| High dependency | 0.8 | 1.40 |
Result: 27% faster trust erosion in high-dependency relationships
Initial Conditions
| Parameter | Symbol | Recommended | Range | Description |
|---|---|---|---|---|
| Initial Trust | $T^0_{ij}$ | 0.50 | [0.0, 1.0] | Starting trust level |
| Initial Reputation | $R^0_{ij}$ | 0.00 | [0.0, 1.0] | Starting reputation damage |
| Baseline Action | $a^{\text{baseline}}$ | Context | $[0, e]$ | Expected cooperation level |
Initial Trust Guidance:
| Relationship History | $T^0$ | $R^0$ | Scenario |
|---|---|---|---|
| First interaction | 0.50 | 0.00 | Neutral starting point |
| Positive reputation | 0.70 | 0.00 | Known reliable partner |
| Prior relationship | 0.60-0.80 | 0.00 | Successful past collaboration |
| Recovery scenario | 0.30 | 0.40 | Post-violation situation |
| Hostile history | 0.20 | 0.60 | Prior conflicts |
Environment-Specific Parameters
TrustDilemma-v0
default_params = {
'n_agents': 2,
'max_steps': 100,
'endowment': 100.0,
'theta': 20.0,
'gamma': 0.65,
'lambda_plus': 0.10,
'lambda_minus': 0.30,
'mu_R': 0.60,
'delta_R': 0.03,
'xi': 0.50,
'kappa': 1.0,
'T_init': 0.50,
'R_init': 0.00,
}
PlatformEcosystem-v0
default_params = {
'n_agents': 5, # 1 platform + 4 developers
'max_steps': 100,
'endowment': [200.0, 100.0, 100.0, 100.0, 100.0], # Platform has more
'theta': 20.0,
'gamma': 0.70, # Higher complementarity in ecosystems
'lambda_plus': 0.10,
'lambda_minus': 0.30,
'mu_R': 0.55, # Slightly lower damage (more forgiving)
'delta_R': 0.04, # Slightly faster decay
'xi': 0.40, # Lower amplification
'kappa': 1.0,
}
SLCD-v0 (Validated Case Study)
# Parameters calibrated to Samsung-Sony S-LCD (2004-2011)
validated_params = {
'n_agents': 2,
'max_steps': 8, # 8 years
'endowment': [100.0, 100.0],
'theta': 20.0, # Validated
'gamma': 0.65, # Validated
'alpha': [0.50, 0.50], # Equal bargaining power
'D_matrix': [[0.0, 0.45],
[0.40, 0.0]], # Moderate mutual dependency
'lambda_plus': 0.10,
'lambda_minus': 0.30,
'T_init': 0.50,
'R_init': 0.00,
}
RenaultNissan-v0 (Validated Case Study)
# Parameters calibrated to Renault-Nissan Alliance (1999-2025)
validated_params = {
'n_agents': 2,
'max_steps': 26, # 26 years
'phases': 5, # Crisis, Recovery, Growth, Ghosn, Post-Ghosn
'theta': 20.0,
'gamma': 0.60,
'lambda_plus': 0.08, # Slower trust building (cross-cultural)
'lambda_minus': 0.32, # Standard erosion
'mu_R': 0.65, # Higher damage (visible scandals)
'delta_R': 0.02, # Slower forgetting
'T_init': 0.30, # Started in crisis
'R_init': 0.20, # Some initial reputation damage
}
Parameter Sensitivity Analysis
High-Impact Parameters
Parameters with largest effect on environment dynamics:
| Parameter | Sensitivity | Impact |
|---|---|---|
| $\lambda^-/\lambda^+$ ratio | Very High | Determines trust recovery feasibility |
| $\gamma$ (complementarity) | High | Controls cooperative incentive strength |
| $D_{ij}$ (interdependence) | High | Shapes utility landscape |
| $\mu_R$ (damage severity) | Medium-High | Determines hysteresis strength |
Low-Impact Parameters
Parameters that can be approximated without significant effect:
| Parameter | Sensitivity | Notes |
|---|---|---|
| $\kappa$ (signal sensitivity) | Low | 1.0 works for most scenarios |
| $\delta_R$ (reputation decay) | Low | Affects long-horizon only |
| $\theta$ vs $\beta$ (specification) | Low within spec | Both work, $\theta$ slightly better validated |
Sensitivity Recommendations
# For robustness testing, vary these parameters:
sensitivity_ranges = {
'lambda_ratio': [2.5, 3.0, 3.5, 4.0], # Primary sensitivity
'gamma': [0.55, 0.65, 0.75], # Secondary sensitivity
'mu_R': [0.50, 0.60, 0.70], # Tertiary sensitivity
}
# Keep these fixed at validated values:
fixed_params = {
'theta': 20.0,
'kappa': 1.0,
'delta_R': 0.03,
}
Calibration Workflow
For New Scenarios
- Start with defaults: Use recommended values from this document
- Identify analogous case: Find most similar validated environment
- Adjust key parameters: Focus on high-impact parameters
- Validate qualitatively: Check behavior matches domain expectations
- Run sensitivity analysis: Test robustness across parameter ranges
For Research Extensions
- Define hypothesis: What parameter relationship are you testing?
- Design sweep: Systematic variation of parameters
- Measure outcomes: Return, trust, cooperation patterns
- Report ranges: Document which parameter ranges support findings
Example: Custom Manufacturing Partnership
# Step 1: Start with SLCD as base (validated manufacturing JV)
params = slcd_params.copy()
# Step 2: Adjust for specific context
params['D_matrix'] = [[0.0, 0.60], # Higher mutual dependency
[0.55, 0.0]]
params['gamma'] = 0.70 # Higher complementarity (specialized assets)
params['T_init'] = 0.65 # Prior positive relationship
# Step 3: Run and validate
env = coopetition_gym.make("TrustDilemma-v0", **params)
# ... run episodes and verify reasonable dynamics
Benchmark-Derived Insights
From 760 experiments across 20 algorithms:
Parameter Combinations That Work Well
| Configuration | Result | Insight |
|---|---|---|
| $\lambda^-/\lambda^+ = 3.0$, $T_{\text{init}} = 0.50$ | Trust-Return r=0.552 | Standard validated setup |
| High $\gamma$ + High $D$ | Cooperation emergence | Strong incentives align behavior |
| Moderate all parameters | Robust performance | Avoids edge case instabilities |
Parameter Combinations That Fail
| Configuration | Result | Insight |
|---|---|---|
| $\lambda^-/\lambda^+ > 5.0$ | Trust collapse | Recovery becomes impossible |
| $\gamma < 0.3$ | Defection dominance | Insufficient cooperative incentive |
| $D_{ij} \to 0$ for all pairs | Pure competition | No structural incentive for cooperation |
Citation
For parameter validation methodology:
@article{pant2025tr1,
title={Computational Foundations for Strategic Coopetition: Formalizing Interdependence and Complementarity},
author={Pant, Vik and Yu, Eric},
journal={arXiv preprint arXiv:2510.18802},
year={2025},
note={Section 7: Validation methodology; 22,000+ trials}
}
@article{pant2025tr2,
title={Computational Foundations for Strategic Coopetition: Formalizing Trust and Reputation Dynamics},
author={Pant, Vik and Yu, Eric},
journal={arXiv preprint arXiv:2510.24909},
year={2025},
note={Section 7: Parameter validation; 78,125 configurations}
}
@article{pant2026tr3,
title={Computational Foundations for Strategic Coopetition: Formalizing Collective Action and Loyalty},
author={Pant, Vik and Yu, Eric},
journal={arXiv preprint arXiv:2601.16237},
year={2026},
note={Loyalty parameters: $\phi_B$=0.8, $\phi_C$=0.3; Apache validation 52/60}
}
@article{pant2026tr4,
title={Computational Foundations for Strategic Coopetition: Formalizing Sequential Interaction and Reciprocity},
author={Pant, Vik and Yu, Eric},
journal={arXiv preprint arXiv:2604.01240},
year={2026},
note={Reciprocity parameters: $\rho_0$, $\eta$, $\kappa$, $k$; Apple App Store validation 48/55}
}
Navigation
- Theoretical Foundations
- Interdependence Framework
- Value Creation & Complementarity
- Trust Dynamics
- Environment Reference
Technical Reports
- TR-1: Computational Foundations for Strategic Coopetition: Formalizing Interdependence and Complementarity (arXiv:2510.18802)
- TR-2: Computational Foundations for Strategic Coopetition: Formalizing Trust and Reputation Dynamics (arXiv:2510.24909)
- TR-3: Computational Foundations for Strategic Coopetition: Formalizing Collective Action and Loyalty (arXiv:2601.16237)
- TR-4: Computational Foundations for Strategic Coopetition: Formalizing Sequential Interaction and Reciprocity (arXiv:2604.01240)