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Visualizing Coalition Formation: From Hedonic Games to Image Segmentation

Pedro Henrique de Paula França, Lucas Lopes Felipe, Daniel Sadoc Menasché

TL;DR

This work links multi-agent systems with image segmentation by quantifying the impact of mechanism design parameters on equilibrium structures by observing transitions from cohesive to fragmented yet recoverable equilibria, and finally to intrinsic failure under excessive fragmentation.

Abstract

We propose image segmentation as a visual diagnostic testbed for coalition formation in hedonic games. Modeling pixels as agents on a graph, we study how a granularization parameter shapes equilibrium fragmentation and boundary structure. On the Weizmann single-object benchmark, we relate multi-coalition equilibria to binary protocols by measuring whether the converged coalitions overlap with a foreground ground-truth. We observe transitions from cohesive to fragmented yet recoverable equilibria, and finally to intrinsic failure under excessive fragmentation. Our core contribution links multi-agent systems with image segmentation by quantifying the impact of mechanism design parameters on equilibrium structures.

Visualizing Coalition Formation: From Hedonic Games to Image Segmentation

TL;DR

This work links multi-agent systems with image segmentation by quantifying the impact of mechanism design parameters on equilibrium structures by observing transitions from cohesive to fragmented yet recoverable equilibria, and finally to intrinsic failure under excessive fragmentation.

Abstract

We propose image segmentation as a visual diagnostic testbed for coalition formation in hedonic games. Modeling pixels as agents on a graph, we study how a granularization parameter shapes equilibrium fragmentation and boundary structure. On the Weizmann single-object benchmark, we relate multi-coalition equilibria to binary protocols by measuring whether the converged coalitions overlap with a foreground ground-truth. We observe transitions from cohesive to fragmented yet recoverable equilibria, and finally to intrinsic failure under excessive fragmentation. Our core contribution links multi-agent systems with image segmentation by quantifying the impact of mechanism design parameters on equilibrium structures.
Paper Structure (27 sections, 1 theorem, 32 equations, 11 figures, 3 algorithms)

This paper contains 27 sections, 1 theorem, 32 equations, 11 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Hedonic Mechanism for Coalition Formation
  3. Results
  4. Conclusion
  5. Pixel-Graph Construction
  6. Numerical Report
  7. Fragmentation Diagnostics
  8. Qualitative Extremes: Peak and Decay Cases
  9. Peak case
  10. Decay case
  11. Sensitivity to Initialization and to Ground Truth Labels
  12. Details of Binary Projections and $F_1$ Computation
  13. F1-single (dominant coalition).
  14. F1-union (recoverable union).
  15. Binary Masks and Pixel Sets
  16. ...and 12 more sections

Key Result

Proposition 1

For any integer $M\ge 1$, there exists a partition $\Pi=\{C_1,\dots,C_M\}$ and a binary ground-truth mask $Y$ such that Consequently, which can be made arbitrarily close to $1$ by taking $M$ large.

Figures (11)

  • Figure 1: Segmentation of the same image for increasing values of $\gamma$. (a) Original image. (b) $\gamma=10^{-6}$. (c) $\gamma=8\times10^{-6}$. (d) $\gamma=10^{-5}$. (e) $\gamma=5\times10^{-4}$. (f) $\gamma=2.5\times10^{-1}$.
  • Figure 2: Pipeline: (a) image $\rightarrow$ (b) graph construction (downsampled,1 see Appendix \ref{['sec:graph']}) $\rightarrow$ (c) equilibrium partition (Section \ref{['sec:mechanism']}) $\rightarrow$ (d) segmented image $\rightarrow$ (e) ground truth evaluation (Section \ref{['sec:results']}).
  • Figure 3: Projections from a multi-community partition. (a) Original image. (b) Binary ground-truth mask. (c) Equilibrium partition by hedonic mechanism. (d) Best single community selected by $\mathrm{F}_1^{\text{single}}$ at $\gamma=7.63\times10^{-6}$. (e) Best subset of communities selected by $\mathrm{F}_1^{\text{union}}$ at $\gamma=2.96\times10^{-5}$.
  • Figure 4: (a) Mean $\mathrm{F}_1^{\text{single}}$ and $\mathrm{F}_1^{\text{union}}$ as a function of $\gamma$ (induced by $\mathrm{density}(G)/900$). (b) Global distributions of $\mathrm{F}_1^{\text{single}}$ and $\mathrm{F}_1^{\text{union}}$ over 100 images after selecting the best GT per image.
  • Figure 5: Diagnostic pipeline: image $\rightarrow$ graph $\rightarrow$ hedonic partition $\rightarrow$ binary projection and evaluation.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 1: Arbitrarily small $\mathrm{F}_1^{\text{single}}$ with $\mathrm{F}_1^{\text{union}}=1$
  • proof : Proof sketch