Eur. Phys. J. B 71, 597-610 (2009)
https://doi.org/10.1140/epjb/e2009-00325-x

Cycles of cooperation and free-riding in social systems

¹ Laboratoire TIMC-IMAG (UMR 5525 CNRS-UJF-INPG), Domaine de la Merci, Jean Roget, 38710 La Tronche, France
² Centre d'Analyse et Mathématique Sociales (UMR 8557 CNRS-EHESS), École des Hautes Études en Sciences Sociales, Paris, France
³ Laboratoire de Physique Statistique (UMR 8550 CNRS-ENS-Paris 6-Paris 7), École Normale Supérieure, Paris, France

Corresponding author: ^a mirta.gordon@imag.fr

Received: 2 January 2009
Revised: 7 July 2009
Published online: 7 October 2009

Abstract

Basic evidences on non-profit making and other forms of benevolent-based organizations reveal a rough partition of members between some pure consumers of the public good (free-riders) and benevolent individuals (cooperators). We study the relationship between the community size and the level of cooperation in a simple model where the utility of joining the community is proportional to its size. We assume an idiosyncratic willingness to join the community ; cooperation bears a fixed cost while free-riding bears a (moral) idiosyncratic cost proportional to the fraction of cooperators. We show that the system presents two types of equilibria: fixed points (Nash equilibria) with a mixture of cooperators and free-riders and cycles where the size of the community, as well as the proportion of cooperators and free-riders, vary periodically.