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On Altruism and Spite in Bimatrix Games

Michail Fasoulakis, Leonidas Bakopoulos, Charilaos Akasiadis, Georgios Chalkiadakis

TL;DR

This work initiates an algorithmic study of altruism and spite in bimatrix games by introducing a parametric transformation $G'=(R',C')$ with $R'=R+\\lambda_R C$ and $C'=C+\\lambda_C R$, enabling analysis of equilibrium complexity and learning dynamics. It establishes PPAD-completeness for exact Nash equilibria in most settings while identifying polynomial-time approximate NE regimes for certain $\\lambda$ values, and presents a gradient-descent framework that alternates strategy and behavioral-parameter optimization to reach low regret. The paper also demonstrates opponent modeling to infer an opponent's altruism/spite level, enabling informed opponent selection and transfer learning across different games to accelerate adaptation. Together, these results bridge algorithmic game theory with machine learning methods to study, learn, and exploit robust altruistic or spiteful behaviors in strategic interactions.

Abstract

One common assumption in game theory is that any player optimizes a utility function that takes into account only its own payoff. However, it has long been observed that in real life players may adopt an altruistic or even spiteful behaviour. As such, there are numerous attempts in the economics literature that strive to explain the fact that players are not entirely selfish, but most of these works do not focus on the algorithmic implications of altruism or spite in games. In this paper, we relax the aforementioned ``self-interest'' assumption, and initiate the study of algorithmic aspects of bimatrix games -- such as the complexity and the quality of their (approximate) Nash equilibria -- under altruism or spite. We provide both a theoretical and an experimental treatment of these topics. Moreover, we demonstrate the potential for learning the degree of an opponent's altruistic/spiteful behaviour, and employing this for opponent selection and transfer of knowledge in bimatrix games.

On Altruism and Spite in Bimatrix Games

TL;DR

This work initiates an algorithmic study of altruism and spite in bimatrix games by introducing a parametric transformation with and , enabling analysis of equilibrium complexity and learning dynamics. It establishes PPAD-completeness for exact Nash equilibria in most settings while identifying polynomial-time approximate NE regimes for certain values, and presents a gradient-descent framework that alternates strategy and behavioral-parameter optimization to reach low regret. The paper also demonstrates opponent modeling to infer an opponent's altruism/spite level, enabling informed opponent selection and transfer learning across different games to accelerate adaptation. Together, these results bridge algorithmic game theory with machine learning methods to study, learn, and exploit robust altruistic or spiteful behaviors in strategic interactions.

Abstract

One common assumption in game theory is that any player optimizes a utility function that takes into account only its own payoff. However, it has long been observed that in real life players may adopt an altruistic or even spiteful behaviour. As such, there are numerous attempts in the economics literature that strive to explain the fact that players are not entirely selfish, but most of these works do not focus on the algorithmic implications of altruism or spite in games. In this paper, we relax the aforementioned ``self-interest'' assumption, and initiate the study of algorithmic aspects of bimatrix games -- such as the complexity and the quality of their (approximate) Nash equilibria -- under altruism or spite. We provide both a theoretical and an experimental treatment of these topics. Moreover, we demonstrate the potential for learning the degree of an opponent's altruistic/spiteful behaviour, and employing this for opponent selection and transfer of knowledge in bimatrix games.

Paper Structure

This paper contains 16 sections, 5 theorems, 27 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic game-theoretic concepts.
  4. Altruism and spite.
  5. Game theoretic aspects of altruism and spite
  6. Computational complexity of computing a Nash Equilibrium under altruism or spite
  7. Regret under altruism or spite
  8. Our Algorithmic framework.
  9. Analysis of Algorithm \ref{['prob:ts_problem']}.
  10. Experimental Evaluation
  11. Approximate NE under altruism and spite
  12. Opponent Modeling and Opponent Selection
  13. Transfer of knowledge and transfer learning
  14. Related Work
  15. Conclusions and future work
  16. ...and 1 more sections

Key Result

theorem thmcountertheorem

For any $\lambda_R\in (-1,1)$ and any $\lambda_C\in (-1,1)$, the problem of computing a NE in ${G'}$ is PPAD-complete.

Figures (4)

  • Figure 1: The $\varepsilon$ approximation for PG for the modified game. Results are averages over 20 runs.
  • Figure 2: Approximation for the TS tight example CDHLL23 for the modified game.
  • Figure 3: The $\epsilon$ approximation for our Prisoner's Dilemma (PD) game in its modified form. Averages over $20$ runs.
  • Figure 4: Alg. \ref{['prob:ts_problem']} approximation for the example of Eq. \ref{['game:ts_problem']}.

Theorems & Definitions (13)

  • definition thmcounterdefinition: Nash equilibria (NE)
  • definition thmcounterdefinition: $\varepsilon$-approximate Nash equilibria
  • definition thmcounterdefinition: Altruism MM08
  • definition thmcounterdefinition: Spite
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • ...and 3 more