Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability Under Valuation Updates in Coalition Formation

Fabian Frank, Matija Novaković, René Romen

TL;DR

This work presents a detailed picture of the complexity of finding nearby stable coalition structures in additively separable hedonic games, for both symmetric and non-symmetric valuations, and presents polynomial-time algorithms for contractual Nash stability and contractual individual stability under restricted symmetric valuations.

Abstract

Coalition formation studies how to partition a set of agents into disjoint coalitions under consideration of their preferences. We study the classical objective of stability in a variant of additively separable hedonic games where agents can change their valuations. Our objective is to find a stable partition after each change. To minimize the reconfiguration cost, we search for nearby stable coalition structures. Our focus is on stability concepts based on single-agent deviations. We present a detailed picture of the complexity of finding nearby stable coalition structures in additively separable hedonic games, for both symmetric and non-symmetric valuations. Our results show that the problem is NP-complete for Nash stability, individual stability, contractual Nash stability, and contractual individual stability. We complement these results by presenting polynomial-time algorithms for contractual Nash stability and contractual individual stability under restricted symmetric valuations. Finally, we show that these algorithms guarantee a bounded average distance over long sequences of updates.

Stability Under Valuation Updates in Coalition Formation

TL;DR

This work presents a detailed picture of the complexity of finding nearby stable coalition structures in additively separable hedonic games, for both symmetric and non-symmetric valuations, and presents polynomial-time algorithms for contractual Nash stability and contractual individual stability under restricted symmetric valuations.

Abstract

Coalition formation studies how to partition a set of agents into disjoint coalitions under consideration of their preferences. We study the classical objective of stability in a variant of additively separable hedonic games where agents can change their valuations. Our objective is to find a stable partition after each change. To minimize the reconfiguration cost, we search for nearby stable coalition structures. Our focus is on stability concepts based on single-agent deviations. We present a detailed picture of the complexity of finding nearby stable coalition structures in additively separable hedonic games, for both symmetric and non-symmetric valuations. Our results show that the problem is NP-complete for Nash stability, individual stability, contractual Nash stability, and contractual individual stability. We complement these results by presenting polynomial-time algorithms for contractual Nash stability and contractual individual stability under restricted symmetric valuations. Finally, we show that these algorithms guarantee a bounded average distance over long sequences of updates.
Paper Structure (18 sections, 41 theorems, 15 equations, 13 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 41 theorems, 15 equations, 13 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Model
  5. Hedonic Games
  6. Stability
  7. Problem Statement
  8. Results
  9. Strict ASHGs
  10. CNS and CIS
  11. NS and IS
  12. Average Distance of Multiple Updates
  13. Conclusion
  14. Missing Proofs in Section \ref{['sec:Results']}
  15. Missing Proofs in Section \ref{['sec:StrictASHGs']}
  16. ...and 3 more sections

Key Result

Lemma 1

The distance $d(\pi, \pi')$ between two partitions $\pi$ and $\pi'$ can be computed in polynomial time. Moreover, checking whether $\pi$ is $X$ for $X \in \{\text{NS, IS, CNS, CIS}\}$ can be done in polynomial time.

Figures (13)

  • Figure 1: Illustration of the symmetric ASHG $G$ from the reduction in \ref{['symmetric_Single_Hardness']}. Dotted rectangles are independent sets. All edges that are not shown correspond to a valuation of $0$. Coalitions in $\pi$ are highlighted in blue.
  • Figure 2: A symmetric FEG where the closest CNS coalition has distance 4. There is an edge between agents $i$ and $j$ if and only if $v(i, j) = 1$. Otherwise, it holds that $v(i,j) = -1$.
  • Figure 3: An FEG where the closest CNS/CIS partition has distance $n-3$ after altering one valuation of agent $n$ depicted for $n = 6$. There is an arc $(i, j)$ between agents $i$ and $j$ if and only if $v_i(j) = 1$. Otherwise, it holds that $v_i(j) = -1$. The original game is on the left and the altered game on the right. Coalitions in $\pi$ and $\pi'$ are highlighted in blue.
  • Figure 4: Illustration of $G$ in \ref{['IS-UpAndDown']}. Rectangles are cliques with valuations of $1$, and edges without an edge weight correspond to valuations of $1$. Coalitions in $\pi$ are highlighted in blue.
  • Figure 5: Illustration of $G$ from the reduction in \ref{['symmetric_Single_Hardness']}. Dotted rectangles are independent sets. All edges that are not shown correspond to a valuation of $0$. Coalitions in $\pi$ are highlighted in blue.
  • ...and 8 more figures

Theorems & Definitions (71)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof : Proof sketch
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof : Proof sketch
  • Theorem 5
  • ...and 61 more