TeamProduction-v0

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

Overview

TeamProduction-v0 implements a team production game where agents contribute effort to a shared production function. This is the baseline TR-3 environment demonstrating free-rider dynamics without loyalty mechanisms.

The environment tests whether reinforcement learning agents can overcome the free-rider temptation,contributing less effort while benefiting from teammates’ contributions.

MARL Classification

Property Value
Game Type N-player Markov Game (general-sum)
Cooperation Structure Mixed-Motive (team production vs. individual cost)
Observability Full (all state variables observable)
Communication Implicit (through actions only)
Agent Symmetry Symmetric (identical capabilities)
Reward Structure Team share minus individual cost
Action Space Continuous, bounded: $A_i = [0, 50]$
State Dynamics Deterministic
Horizon Finite, T = 100 steps
Canonical Comparison Team production games; Holmström (1982); Alchian & Demsetz (1972)

Formal Specification

Mathematical Framework (TR-3)

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

Where:

Base Payoff: \(\pi_i^{team} = \frac{1}{n} \cdot Q(\mathbf{a}) - c \cdot a_i\)

Where:

Nash Equilibrium (Free-Riding): \(a^* = \left(\frac{\omega\beta}{nc}\right)^{\frac{1}{1-\beta}}\)

Social Optimum: \(a^{opt} = \left(\frac{\omega\beta}{c}\right)^{\frac{1}{1-\beta}}\)

State Space

S ⊆ ℝ^d with components:

Component Symbol Description
Actions a Previous effort levels
Trust Matrix τ Pairwise trust (from TR-2)
Reputation R Accumulated reputation damage
Interdependence D Structural dependencies
Loyalty Scores θ Per-agent loyalty (initialized neutral)

Action Space

For each agent $i$: \(A_i = [0, a_{max}] = [0, 50] \subset \mathbb{R}\)

Actions represent effort contribution to team production.

Uniaxial Treatment: This environment uses the single-dimension action space characteristic of Coopetition-Gym v1.x. The free-rider dynamic emerges through divergence between individual and collective incentives rather than explicit competitive actions.

Reward Function

In the baseline TeamProduction-v0, rewards are pure team payoffs without loyalty:

\[r_i = \frac{1}{n} \cdot Q(\mathbf{a}) - c \cdot a_i\]

With light penalty for severe free-riding (cooperation rate < 20%).

Game-Theoretic Background

The Free-Rider Problem

Team production creates a classic collective action problem: 1. Individual incentive: Contribute less effort (save cost) while sharing output

  1. Collective outcome: If all free-ride, team output is minimal
  2. Nash equilibrium: Suboptimal effort level $a^* < a^{opt}$

Strategic Implications

Free-Rider Equilibrium:

Social Optimum:

Price of Anarchy: \(PoA = \frac{\text{Social Welfare at Optimum}}{\text{Social Welfare at Nash}} \approx 2.5\)

Environment Specification

Basic Usage

import coopetition_gym
import numpy as np

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

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

# Check Nash equilibrium
print(f"Nash effort: {info['nash_equilibrium']:.2f}")
print(f"Social optimum: {info['social_optimum']:.2f}")

# Run episode with Nash equilibrium strategy
for step in range(100): nash_effort = info['nash_equilibrium']
    actions = np.array([nash_effort] * 4)
    obs, rewards, terminated, truncated, info = env.step(actions)

    if terminated or truncated: break

print(f"Team output at Nash: {info['team_output']:.2f}")

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
max_steps 100 Maximum timesteps
render_mode None Rendering mode

Spaces

Observation Space

Type: Box Dtype: float32

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

Action Space

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

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
free_rider_count int Agents below threshold
efficiency_ratio float Actual/optimal effort ratio
mean_trust float Average trust level

Key Dynamics

Free-Rider Detection

Agents with cooperation rate below 30% are flagged as free-riders:

free_riders = [i for i in range(n) if actions[i]/endowments[i] < 0.3]

Baseline Behavior

Without loyalty mechanisms:

This establishes the baseline for comparing with LoyaltyTeam-v0.

Research Applications

TeamProduction-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. Holmström, B. (1982). Moral Hazard in Teams. Bell Journal of Economics.
  3. Alchian, A. & Demsetz, H. (1972). Production, Information Costs, and Economic Organization. American Economic Review.