LoyaltyTeam-v0

Category: Collective Action Environment (TR-3) Agents: 4 (configurable) Difficulty: Intermediate-Advanced Source: coopetition_gym/envs/collective_action_envs.py

Overview

LoyaltyTeam-v0 implements team production with full TR-3 loyalty mechanisms. Unlike the baseline TeamProduction-v0, agents with high loyalty receive bonuses proportional to teammate welfare, creating positive-sum dynamics that can sustain cooperation above the Nash equilibrium.

This environment tests whether loyalty dynamics can overcome the free-rider problem and sustain cooperation in team settings.

MARL Classification

Property Value
Game Type N-player Markov Game (general-sum)
Cooperation Structure Mixed-Motive with loyalty amplification
Observability Full
Communication Implicit (through actions)
Agent Symmetry Symmetric (capabilities), asymmetric (loyalty states)
Reward Structure Team share + loyalty modifier
Action Space Continuous: $A_i = [0, 50]$
State Dynamics Deterministic with loyalty evolution
Horizon Finite, T = 100 steps
Canonical Comparison Behavioral team theory; TR-3 loyalty model

Formal Specification

Mathematical Framework (TR-3)

Team Production Function: \(Q(\mathbf{a}) = \omega \cdot \left(\sum_{i=1}^{n} a_i\right)^\beta\)

Loyalty Modifier: \(L_i = \theta_i \cdot \left[\phi_B \cdot \bar{\pi}_{-i} + \phi_C \cdot c \cdot a_i\right]\)

Where:

Loyalty-Augmented Utility: \(U_i = \pi_i^{team} + L_i\)

Loyalty Evolution

Loyalty scores evolve based on cooperation:

if cooperation_rate >= 0.5: loyalty += 0.02 * cooperation_rate  # Build slowly
else: loyalty -= 0.05 * (0.5 - cooperation_rate)  # Erode faster

Key Insight

High-loyalty agents ($\theta_i \approx 1$) receive substantial bonuses when:

  1. Teammates are doing well ($\phi_B \cdot \bar{\pi}_{-i}$)
  2. They contribute high effort themselves ($\phi_C \cdot c \cdot a_i$)

This creates intrinsic motivation for sustained cooperation.

Environment Specification

Basic Usage

import coopetition_gym
import numpy as np

# Create environment with default loyalty parameters
env = coopetition_gym.make("LoyaltyTeam-v0")

# Or customize loyalty strengths
env = coopetition_gym.make(
    "LoyaltyTeam-v0",
    phi_B=0.9,  # Strong teammate-welfare bonus
    phi_C=0.4,  # Moderate cost tolerance
)

obs, info = env.reset(seed=42)

# Sustained cooperation builds loyalty
for step in range(50):
    # High cooperation
    actions = np.array([40.0, 40.0, 40.0, 40.0])
    obs, rewards, terminated, truncated, info = env.step(actions)

print(f"Mean loyalty after 50 steps: {info['mean_loyalty']:.2f}")
print(f"Loyalty lift over Nash: {info['loyalty_lift']:.2f}x")

Parameters

Parameter Default Description
n_agents 4 Number of team members
omega 25.0 Productivity factor
beta 0.7 Returns to scale
c 1.0 Effort cost coefficient
phi_B 0.8 Loyalty benefit strength
phi_C 0.3 Cost tolerance strength
max_steps 100 Maximum timesteps
render_mode None Rendering mode

Spaces

Observation Space

Type: Box Dtype: float32

Includes actions, trust matrix, reputation, interdependence, loyalty scores, and step info.

Action Space

Type: Box Shape: (n_agents,) Dtype: float32 Range: [0.0, 50.0] for each agent

Uniaxial Treatment: This environment uses the single-dimension action space characteristic of Coopetition-Gym v1.x. Competition emerges through the free-rider dynamic modulated by loyalty mechanisms rather than explicit competitive actions.

Metrics and Info

The info dictionary contains:

Key Type Description
step int Current timestep
team_output float Q(a) = ω·(Σaᵢ)^β
nash_equilibrium float Theoretical Nash effort
social_optimum float Theoretical optimal effort
mean_loyalty float Average loyalty score
loyalty_scores list Per-agent loyalty [0,1]
loyalty_lift float Output ratio vs. Nash baseline
team_cohesion float Weighted loyalty metric
free_rider_count int Agents below threshold
efficiency_ratio float Actual/optimal effort ratio

Key Dynamics

Loyalty Building

Sustained cooperation builds loyalty over time:

Behavior Loyalty Change
High cooperation (>50%) +0.02 × rate
Low cooperation (<50%) -0.05 × deficit
Sustained high Approaches 1.0
Sustained low Approaches 0.0

Loyalty Lift

The key metric measuring TR-3 effectiveness:

\[\text{Loyalty Lift} = \frac{\text{Actual Team Output}}{\text{Nash Equilibrium Output}}\]

With effective loyalty mechanisms, teams can achieve lift > 2.0.

Phase Dynamics

  1. Early phase (steps 1-20): Loyalty neutral (~0.5), behavior determines trajectory
  2. Middle phase (steps 20-60): Loyalty diverges based on cooperation patterns
  3. Late phase (steps 60-100): Loyalty stable, determines final outcomes

Comparison with TeamProduction-v0

Aspect TeamProduction-v0 LoyaltyTeam-v0
Loyalty mechanisms None Full TR-3
Expected equilibrium Nash (low) Above Nash possible
Free-rider penalty Light Moderate + loyalty erosion
Cooperation incentive External only Intrinsic (loyalty bonus)
Typical output ~baseline 1.5-2.5× baseline

Research Applications

LoyaltyTeam-v0 is suitable for studying:

Example: Loyalty Evolution Tracking

import coopetition_gym
import numpy as np

env = coopetition_gym.make("LoyaltyTeam-v0")
obs, info = env.reset(seed=42)

loyalty_history = []

for step in range(100):
    # Cooperative strategy
    actions = np.array([35.0, 35.0, 35.0, 35.0])
    obs, rewards, terminated, truncated, info = env.step(actions)

    loyalty_history.append(info['mean_loyalty'])

    if terminated or truncated: break

# Loyalty should increase over time with cooperation
print(f"Initial loyalty: {loyalty_history[0]:.2f}")
print(f"Final loyalty: {loyalty_history[-1]:.2f}")
print(f"Loyalty lift: {info['loyalty_lift']:.2f}x")

References

  1. Pant, V. & Yu, E. (2026). Computational Foundations for Strategic Coopetition: Formalizing Collective Action and Loyalty. arXiv:2601.16237
  2. Akerlof, G. & Kranton, R. (2010). Identity Economics. Princeton University Press.
  3. Kandel, E. & Lazear, E. (1992). Peer Pressure and Partnerships. Journal of Political Economy.