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The Complexity of Manipulation of k-Coalitional Games on Graphs

Hodaya Barr, Yohai Trabelsi, Sarit Kraus, Liam Roditty, Noam Hazon

TL;DR

This work analyzes the computational complexity of manipulating partitions in k-coalitional graphs where agent utilities are measured by intra-coalition friendships. It introduces a novel socially-aware manipulation (SAM) paradigm and studies manipulation and improvement problems across three organizer objectives (Max-Egal, At-Least-1, Max-Util), deriving hardness results (co-NP/NP) and providing an XP-algorithm for Max-Util that can compute optimal manipulations, with polynomial-time SAM solutions when k is fixed. The authors connect manipulation dynamics to min-cut and min-k-cut structures, enabling tractable strategies for exact and approximate solutions, and they validate SAM’s frequency and the XP approach via simulations on social-network-derived graphs. The study highlights the practical significance of SAM and outlines open questions, including exact complexity for Max-Util manipulations and multi-manipulator extensions. Overall, the paper advances the theory of manipulation in cooperative graph games and offers practical algorithms for socially aware strategic behavior.

Abstract

In many settings, there is an organizer who would like to divide a set of agents into $k$ coalitions, and cares about the friendships within each coalition. Specifically, the organizer might want to maximize utilitarian social welfare, maximize egalitarian social welfare, or simply guarantee that every agent will have at least one friend within his coalition. However, in many situations, the organizer is not familiar with the friendship connections, and he needs to obtain them from the agents. In this setting, a manipulative agent may falsely report friendship connections in order to increase his utility. In this paper, we analyze the complexity of finding manipulation in such $k$-coalitional games on graphs. We also introduce a new type of manipulation, socially-aware manipulation, in which the manipulator would like to increase his utility without decreasing the social welfare. We then study the complexity of finding socially-aware manipulation in our setting. Finally, we examine the frequency of socially-aware manipulation and the running time of our algorithms via simulation results.

The Complexity of Manipulation of k-Coalitional Games on Graphs

TL;DR

This work analyzes the computational complexity of manipulating partitions in k-coalitional graphs where agent utilities are measured by intra-coalition friendships. It introduces a novel socially-aware manipulation (SAM) paradigm and studies manipulation and improvement problems across three organizer objectives (Max-Egal, At-Least-1, Max-Util), deriving hardness results (co-NP/NP) and providing an XP-algorithm for Max-Util that can compute optimal manipulations, with polynomial-time SAM solutions when k is fixed. The authors connect manipulation dynamics to min-cut and min-k-cut structures, enabling tractable strategies for exact and approximate solutions, and they validate SAM’s frequency and the XP approach via simulations on social-network-derived graphs. The study highlights the practical significance of SAM and outlines open questions, including exact complexity for Max-Util manipulations and multi-manipulator extensions. Overall, the paper advances the theory of manipulation in cooperative graph games and offers practical algorithms for socially aware strategic behavior.

Abstract

In many settings, there is an organizer who would like to divide a set of agents into coalitions, and cares about the friendships within each coalition. Specifically, the organizer might want to maximize utilitarian social welfare, maximize egalitarian social welfare, or simply guarantee that every agent will have at least one friend within his coalition. However, in many situations, the organizer is not familiar with the friendship connections, and he needs to obtain them from the agents. In this setting, a manipulative agent may falsely report friendship connections in order to increase his utility. In this paper, we analyze the complexity of finding manipulation in such -coalitional games on graphs. We also introduce a new type of manipulation, socially-aware manipulation, in which the manipulator would like to increase his utility without decreasing the social welfare. We then study the complexity of finding socially-aware manipulation in our setting. Finally, we examine the frequency of socially-aware manipulation and the running time of our algorithms via simulation results.
Paper Structure (27 sections, 26 theorems, 1 equation, 9 figures, 3 tables, 4 algorithms)

This paper contains 27 sections, 26 theorems, 1 equation, 9 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Preliminaries
  2. Background
  3. The Complexity of the Manipulation and the Improvement Problems
  4. Socially-aware Manipulation
  5. The Existence of SAM
  6. The Complexity of the Manipulation and Improvement Problems for SAM
  7. Experiments
  8. Frequency of SAM.
  9. Effectiveness of the XP algorithm.
  10. Conclusions and Future Work
  11. Omitted Proofs
  12. Proof of Theorem \ref{['thm:sim_me_directed_hard']}
  13. Proof of Theorem \ref{['thm:improvment_me_directed_hard']}
  14. Proof of Theorem \ref{['thm:alo_hard_compute']}
  15. Proof of Theorem \ref{['thm:improvment_alo_directed_hard']}
  16. ...and 12 more sections

Key Result

Theorem 1

Given a graph $G$, and a manipulator $m^- \in A$, deciding whether any type of manipulation exists when the objective is Max-Egal is co-$NP$-hard.

Figures (9)

  • Figure 1: An illustration of the four types of manipulations.
  • Figure 2: A switch
  • Figure 3: (a) An illustration of a ring graph for $\mathcal{F} = (x_1 \lor \bar{x_2} \lor x_3) \land (\bar{x_1} \lor x_2 \lor x_3) \land (x_1 \lor \bar{x_2} \lor \bar{x_3})$ (b) The ring graph for the same $\mathcal{F}$ with the additions of clause vertices.
  • Figure 4: Graphs for Theorem \ref{['thm:exist-no-harm-man']}. A dashed line is an edge that is added by $m$. A dotted line is an edge that is removed by $m$. A square represents a clique of size $10$, and an edge with a number represents a set of edges.
  • Figure 5: Average running time (seconds).
  • ...and 4 more figures

Theorems & Definitions (75)

  • Definition 1: Max-Egal
  • Definition 2: At-Least-1
  • Definition 3: Max-Util
  • Definition 4: $m^+$
  • Definition 5: $m^-$
  • Definition 6: Lower Bound Manipulation (LBM)
  • Definition 7: Upper Bound Manipulation (UBM)
  • Definition 8: Weak-Improvement Manipulation (WIM)
  • Definition 9: Strict-Improvement Manipulation (SIM)
  • Definition 10: The Manipulation Problem
  • ...and 65 more