Attraction of the core and the cohesion flow

TL;DR

This paper models the evolution of cooperative game states using a continuous-time dynamical system, revealing that the core acts as a unique attractor for balanced games, thus deepening understanding of core stability.

Contribution

It introduces a novel dynamical system approach to analyze the core and cohesion flow in cooperative games, establishing the core as a unique attractor for balanced games.

Findings

The core is the unique minimal attractor of the cohesion flow.

Each preimputation has a unique associated cohesion curve.

The approach enhances understanding of core stability in cooperative games.

Abstract

We adopt a continuous-time dynamical system approach to study the evolution of the state of a game driven by the willingness to reduce the total dissatisfaction of the coalitions about their payment. Inspired by the work of Grabisch and Sudh\"olter about core stability, we define a vector field on the set of preimputations from which is defined, for any preimputation, a cohesion curve describing the evolution of the state. We prove that for each preimputation, there exists a unique cohesion curve. Subsequently, we show that, for the cohesion flow of a balanced game, the core is the unique minimal attractor of the flow, the realm of which is the whole preimputation set. These results improve our understanding of the ubiquity of the core in the study of cooperative games with transferable utility.