PublicGoods-v0

Category: Collective Action Environment (TR-3) Agents: 5 (configurable) Difficulty: Intermediate Source: coopetition_gym/envs/collective_action_envs.py

Overview

PublicGoods-v0 implements a classic public goods game with TR-3 collective action modifiers. Agents contribute to a public good that benefits everyone equally, regardless of individual contribution levels.

This environment provides a well-studied baseline for testing collective action mechanisms against classic game theory predictions.

MARL Classification

Property Value
Game Type N-player Markov Game (symmetric)
Cooperation Structure Pure public goods with non-excludability
Observability Full
Communication Implicit
Agent Symmetry Symmetric
Reward Structure Kept endowment + share of public good
Action Space Continuous: $A_i = [0, \text{endowment}]$
Horizon Finite, T = 100 steps
Canonical Comparison Linear public goods game; Isaac & Walker (1988)

Formal Specification

Public Goods Mechanism

Contribution: Each agent has endowment $e$ and contributes $a_i \in [0, e]$ to the public good.

Public Good Value: \(G = m \cdot \sum_{i=1}^{n} a_i\)

Where $m > 1$ is the multiplier (making cooperation socially beneficial).

Payoff: \(\pi_i = (e - a_i) + \frac{G}{n} = (e - a_i) + \frac{m \cdot \sum_j a_j}{n}\)

Game-Theoretic Analysis

Nash Equilibrium: With $m < n$ (always true for meaningful games), the dominant strategy is: \(a_i^* = 0 \quad \text{(contribute nothing)}\)

Social Optimum: Full contribution maximizes total welfare: \(a_i^{opt} = e\)

Price of Anarchy: \(PoA = \frac{n \cdot e \cdot m}{n \cdot e} = m\)

TR-3 Loyalty Enhancement

With loyalty mechanisms enabled: \(U_i = \pi_i + L_i\)

Where $L_i = \theta_i \cdot [\phi_B \cdot \bar{\pi}_{-i} + \phi_C \cdot a_i]$

This can sustain positive contributions when loyalty is high.

Environment Specification

Basic Usage

import coopetition_gym
import numpy as np

# Create environment
env = coopetition_gym.make("PublicGoods-v0", n_agents=5, multiplier=2.0)

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

# Full cooperation
for step in range(100):
    # Everyone contributes fully
    actions = np.array([50.0] * 5)
    obs, rewards, terminated, truncated, info = env.step(actions)

    if terminated or truncated: break

print(f"Total contribution: {info['total_contribution']:.1f}")
print(f"Public good value: {info['public_good']:.1f}")
print(f"Contribution rate: {info['contribution_rate']:.1%}")
print(f"Social efficiency: {info['social_efficiency']:.1%}")

Parameters

Parameter Default Description
n_agents 5 Number of players
multiplier 2.0 Public good multiplier (m)
endowment 50.0 Each agent’s starting endowment
phi_B 0.6 Loyalty benefit strength
phi_C 0.2 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, endowment] for each agent

Uniaxial Treatment: This environment uses the single-dimension action space characteristic of Coopetition-Gym v1.x. Competition emerges through the classic public goods free-rider tension rather than explicit competitive actions.

Metrics and Info

The info dictionary contains:

Key Type Description
step int Current timestep
total_contribution float Sum of all contributions
public_good float m × total_contribution
contribution_rate float Total / (n × endowment)
social_efficiency float Actual / maximum welfare
mean_loyalty float Average loyalty score
team_cohesion float Weighted loyalty metric
free_rider_count int Agents contributing below threshold

Key Dynamics

The Public Goods Dilemma

Individual incentive:

Collective benefit:

Loyalty-Mediated Cooperation

TR-3 loyalty mechanisms can overcome the dilemma: 1. High loyalty ($\theta \approx 1$):

  1. Low loyalty ($\theta \approx 0$):
    • Reverts to standard Nash (zero contribution)
    • No intrinsic cooperation motivation

Example: Varying Multiplier Effects

import coopetition_gym
import numpy as np

multipliers = [1.5, 2.0, 3.0, 4.0]
results = {}

for m in multipliers: env = coopetition_gym.make("PublicGoods-v0", multiplier=m)
    obs, info = env.reset(seed=42)

    # Mixed contribution strategy
    for step in range(50): actions = np.array([40.0] * 5)  # 80% contribution
        obs, rewards, terminated, truncated, info = env.step(actions)
        if terminated or truncated: break

    results[m] = {
        "public_good": info['public_good'],
        "efficiency": info['social_efficiency'],
        "mean_loyalty": info['mean_loyalty'],
    }
    env.close()

for m, data in results.items(): print(f"Multiplier {m}: PG={data['public_good']:.1f}, Eff={data['efficiency']:.1%}")

Research Applications

PublicGoods-v0 is suitable for studying:

Literature Connection

Experimental Economics

The public goods game is one of the most studied experimental economics paradigms:

Finding Reference PublicGoods-v0 Relevance
~50% contribution Isaac & Walker (1988) Baseline comparison
Decay over time Ledyard (1995) Loyalty dynamics
Punishment helps Fehr & Gächter (2000) Exclusion mechanics
Communication helps Ostrom (2000) Trust mechanisms

Connection to TR-3

PublicGoods-v0 bridges experimental findings with TR-3 theory:

References

  1. Pant, V. & Yu, E. (2026). Computational Foundations for Strategic Coopetition: Formalizing Collective Action and Loyalty. arXiv:2601.16237
  2. Isaac, R. M. & Walker, J. M. (1988). Group Size Effects in Public Goods Provision: The Voluntary Contributions Mechanism. Quarterly Journal of Economics.
  3. Ledyard, J. (1995). Public Goods: A Survey of Experimental Research. Handbook of Experimental Economics.
  4. Fehr, E. & Gächter, S. (2000). Cooperation and Punishment in Public Goods Experiments. American Economic Review.