[Paper Review] Cooperation in Subset Team Games: Altruism and Selfishness

Elisha Peterson|ArXiv.org|Jul 14, 2009

Evolutionary Game Theory and Cooperation11 references3 citations

TL;DR

This paper introduces subset team games as a framework to unify individual and group utilities by defining altruism and selfishness through a two-dimensional cooperation space, using subset utility functions. It demonstrates that team stability and fairness are significantly enhanced when payoff schemes align with altruistic contributions, particularly in games like Prisoner’s Dilemma and Cobb-Douglas utility models.

ABSTRACT

This paper extends the theory of subset team games, a generalization of cooperative game theory requiring a payoff function that is defined for all subsets of players. This subset utility is used to define both altruistic and selfish contributions of a player to the team. We investigate properties of these games, and analyze the implications of altruism and selfishness for general situations, for prisoner's dilemma, and for a specific game with a Cobb-Douglas utility.

Motivation & Objective

• To extend cooperative game theory by introducing subset team games that model altruism and selfishness through a two-dimensional cooperation space.
• To address the limitations of classical cooperative game theory in modeling real-world teamwork where individuals prioritize group utility over individual gain.
• To develop metrics for altruistic and selfish contributions that quantify team cohesion and stability.
• To analyze how payoff schemes influence team behavior, particularly in high-stakes or asymmetric cooperation scenarios.
• To demonstrate the framework’s applicability in canonical games like Prisoner’s Dilemma and in quantitative models using Cobb-Douglas utility functions.

Proposed method

• Define a subset utility function $u: 2^T \to \mathbb{R}$ over all coalitions $S \subset T$, generalizing classical TU games.
• Introduce two key metrics: altruistic contribution $a_A = u_A(A \cup B) - u_A(B)$ and selfish contribution $c_A = u_A(A \cup B) - u_A(A)$ for disjoint coalitions $A$ and $B$.
• Construct a cooperation space with coordinates $(a_A, c_A)$, where Quadrant I represents fully cooperative behavior.
• Apply the framework to Prisoner’s Dilemma to analyze how altruism affects defection incentives and team stability.
• Use a Cobb-Douglas utility function $f_{A\cup B}(A\cup B)^\theta \hat{x}_{A\cup B}^{1-\theta}$ to model resource allocation and compute contributions under varying payoff schemes.
• Analyze stability through curves plotting $c_A$ and $a_A$ as functions of $x_B/|B|$, with $\gamma \in [0,1]$ controlling payoff proportionality.

Experimental results

Research questions

• RQ1 How can altruism and selfishness be formally quantified within a cooperative game framework that accounts for all subsets of players?
• RQ2 What conditions ensure team stability in subset team games, particularly when players have conflicting incentives?
• RQ3 How do different payoff schemes—equal vs. proportional—impact the level of altruism and the risk of free-riding in team settings?
• RQ4 In what ways does the cooperation space framework improve upon classical marginal contribution analysis in cooperative game theory?
• RQ5 Can the subset team game model predict and optimize team performance in real-world scenarios like resource allocation or collaborative platforms?

Key findings

• Team stability is highest when team sizes are balanced and payoff schemes are proportional ($\gamma = 1$), especially when $x_B/|B|$ is small.
• The selfish contribution $c_A$ increases with $\gamma$, indicating that proportional payoffs better incentivize rational behavior.
• For small $\gamma$, the marginal benefit of 'cheating' increases significantly as $x_B/|B| \to 1$, revealing vulnerability to free-riding.
• The game is fully-cooperative if and only if $f_B(A \cup B) \geq f_B(B)$ for all disjoint $A, B$, ensuring non-negative altruistic contributions.
• The cooperation space framework visualizes team behavior across quadrants, with Quadrant I representing optimal teamwork and Quadrant II reflecting antagonistic, non-cooperative dynamics.
• The framework successfully generalizes classical cooperative game theory by embedding altruism and selfishness into a geometric, analyzable space that predicts team cohesion and incentive compatibility.