CoalitionFormation-v0

Category: Collective Action Environment (TR-3) Agents: 6 (configurable) Difficulty: Advanced Source: coopetition_gym/envs/collective_action_envs.py

Overview

CoalitionFormation-v0 implements dynamic coalition mechanics with entry/exit dynamics. Agents can be excluded from the coalition if their loyalty falls too low, and excluded agents work independently with reduced payoffs.

This environment tests whether agents can maintain coalition stability while managing free-rider threats through exclusion mechanisms.

MARL Classification

Property Value
Game Type N-player Markov Game with coalition structure
Cooperation Structure Coalition-based with exclusion
Observability Full
Communication Implicit (through actions)
Agent Symmetry Symmetric initially, dynamic membership
Reward Structure Coalition share or independent work
Action Space Continuous: $A_i = [0, 50]$
State Dynamics Stochastic coalition transitions
Horizon Finite, T = 150 steps
Early Termination Coalition collapse (< min_size)
Canonical Comparison Coalition formation games; Greenberg (1994)

Formal Specification

Coalition Dynamics

Coalition Membership:

Reward Structure:

For coalition members: \(r_i = \frac{1}{|C|} \cdot Q(\mathbf{a}_C) - c \cdot a_i + \text{stability bonus}\)

For excluded agents: \(r_i = 0.5 \times (\text{independent payoff})\)

Coalition Stability

Minimum Coalition Size: Coalition must maintain at least min_coalition_size members.

Exclusion Mechanism:

if loyalty[i] < exit_threshold: if len(coalition) > min_coalition_size: exclude(agent_i)
        exclusion_timer[i] = reentry_cooldown

Reentry Mechanism:

if i in excluded and timer[i] <= 0: if loyalty[i] >= 0.4: readmit(agent_i)

Environment Specification

Basic Usage

import coopetition_gym
import numpy as np

# Create environment
env = coopetition_gym.make("CoalitionFormation-v0")

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

# Mixed behavior: some cooperate, some free-ride
for step in range(50):
    # 3 cooperators, 3 free-riders
    actions = np.array([40, 40, 40, 5, 5, 5])
    obs, rewards, terminated, truncated, info = env.step(actions)

    if terminated: print("Coalition collapsed!")
        break

print(f"Coalition size: {info['coalition_size']}")
print(f"Excluded agents: {info['excluded_agents']}")

Parameters

Parameter Default Description
n_agents 6 Number of potential members
min_coalition_size 3 Minimum viable coalition
exit_threshold 0.15 Loyalty below which exclusion occurs
reentry_cooldown 5 Steps before excluded agent can rejoin
omega 25.0 Productivity factor
beta 0.7 Returns to scale
max_steps 150 Maximum timesteps
render_mode None Rendering mode

Spaces

Observation Space

Type: Box Dtype: float32

Includes actions, trust, reputation, interdependence, loyalty scores, coalition membership indicators.

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 coalition exclusion dynamics rather than explicit competitive actions.

Metrics and Info

The info dictionary contains:

Key Type Description
step int Current timestep
coalition_size int Current coalition members
coalition_members list Indices of active members
excluded_agents list Indices of excluded agents
coalition_stability float Coalition size / total agents
team_output float Production from coalition
mean_loyalty float Average loyalty score
free_rider_count int Low-contribution agents

Key Dynamics

Exclusion Effects

Condition Effect
Loyalty < 0.15 Risk of exclusion
Coalition > min_size Exclusion occurs
Coalition = min_size Free-riders tolerated
Excluded agent 50% reward penalty

Coalition Stability Bonus

Coalition members receive a 10% bonus when coalition is stable:

if len(coalition) >= min_coalition_size: modifiers[coalition_members] *= 1.1

Termination Condition

Episode terminates if coalition collapses below minimum:

if len(coalition_members) < min_coalition_size: terminated = True

Strategic Considerations

For Coalition Members

For Potential Free-Riders

Emergent Dynamics

Example: Coalition Dynamics Tracking

import coopetition_gym
import numpy as np

env = coopetition_gym.make("CoalitionFormation-v0", n_agents=6)
obs, info = env.reset(seed=42)

coalition_history = []

for step in range(150):
    # Heterogeneous efforts
    actions = np.array([45, 40, 35, 30, 15, 10])
    obs, rewards, terminated, truncated, info = env.step(actions)

    coalition_history.append(info['coalition_size'])

    if terminated: print(f"Terminated at step {step}: coalition collapsed")
        break
    if truncated: break

print(f"Initial coalition: 6")
print(f"Final coalition: {coalition_history[-1]}")
print(f"Excluded agents: {info['excluded_agents']}")

Research Applications

CoalitionFormation-v0 is suitable for studying:

References

  1. Pant, V. & Yu, E. (2026). Computational Foundations for Strategic Coopetition: Formalizing Collective Action and Loyalty. arXiv:2601.16237
  2. Ray, D. (2007). A Game-Theoretic Perspective on Coalition Formation. Oxford University Press.
  3. Greenberg, J. (1994). Coalition Structures. Handbook of Game Theory, Vol. 2.