[Paper Review] Stable partitions in coalitional games

Krzysztof R. Apt, Tadeusz Radzik|ArXiv.org|May 29, 2006

Game Theory and Voting SystemsEconomics, Econometrics and Finance10 references93 citations

TL;DR

This paper introduces a novel notion of stable partitions in coalitional games using a defection function that evaluates alternative coalition arrangements based on social welfare. It characterizes stability under different defection functions and proves that strictly stable partitions exist when a unique partition maximizes social welfare, with applications to transportation cost optimization in store chains.

ABSTRACT

We propose a notion of a stable partition in a coalitional game that is parametrized by the concept of a defection function. This function assigns to each partition of the grand coalition a set of different coalition arrangements for a group of defecting players. The alternatives are compared using their social welfare. We characterize the stability of a partition for a number of most natural defection functions and investigate whether and how so defined stable partitions can be reached from any initial partition by means of simple transformations. The approach is illustrated by analyzing an example in which a set of stores seeks an optimal transportation arrangement.

Motivation & Objective

• To formalize stability in coalitional games by introducing a defection function that evaluates viable alternatives for defecting groups.
• To analyze how different defection functions—such as allowing all partitions of subsets or the grand coalition—affect stability.
• To investigate whether stable partitions can be reached from any initial partition through local transformations.
• To establish existence conditions for strictly stable partitions using social welfare maximization.
• To apply the framework to a real-world example of store transportation arrangements to demonstrate practical relevance.

Proposed method

• Proposes a defection function that maps each partition to a set of alternative coalition arrangements for defecting players.
• Uses social welfare (sum of coalition values) as the criterion to compare alternative arrangements.
• Defines a partition as stable if no defecting group can improve social welfare via any alternative arrangement.
• Characterizes stability under two natural defection functions: one allowing all partitions of subsets, another allowing all partitions of the grand coalition.
• Applies theorems to show that a strictly stable partition exists if and only if a unique partition maximizes social welfare.
• Uses local transformations (splits and merges) to analyze reachability of stable partitions from any initial state.

Experimental results

Research questions

• RQ1 Under what conditions does a strictly stable partition exist in a coalitional game with a defection function?
• RQ2 How do different defection functions affect the stability of a given partition?
• RQ3 Can any stable partition be reached from an arbitrary initial partition through local transformations?
• RQ4 What role does social welfare play in determining the stability of a coalition structure?
• RQ5 How can the proposed framework be applied to real-world problems like transportation cost optimization in store networks?

Key findings

• A strictly $Δ_c$-stable partition exists if and only if exactly one partition maximizes social welfare.
• For strictly superadditive games restricted to subsets of a coalition, the grand coalition ${ N }$ is strictly $Δ_c$-stable.
• In the transportation example, the city-based partition is strictly $Δ_c$-stable due to economies of scale and lower per-store costs within cities.
• The initial chain-based transportation arrangement is unstable and can be transformed into the stable city-based arrangement via a sequence of local splits and merges.
• The existence of a strictly $Δ_p$-stable partition is equivalent to the existence of a unique social welfare-maximizing partition.
• The framework ensures that stable partitions are reachable from any initial partition through simple, local transformations.