Clustering in large networks does not promote upstream reciprocity

TL;DR

This paper demonstrates that the presence of short cycles like triangles in large social networks does not significantly promote upstream reciprocity, and larger populations or scale-free networks tend to reduce cooperation.

Contribution

It extends upstream reciprocity models to general networks, showing that network structure and size have limited impact on cooperation promotion.

Findings

Abundance of triangles does not promote upstream reciprocity.

Larger populations decrease cooperation levels.

Scale-free networks lead to less cooperation than homogeneous networks.

Abstract

Upstream reciprocity (also called generalized reciprocity) is a putative mechanism for cooperation in social dilemma situations with which players help others when they are helped by somebody else. It is a type of indirect reciprocity. Although upstream reciprocity is often observed in experiments, most theories suggest that it is operative only when players form short cycles such as triangles, implying a small population size, or when it is combined with other mechanisms that promote cooperation on their own. An expectation is that real social networks, which are known to be full of triangles and other short cycles, may accommodate upstream reciprocity. In this study, I extend the upstream reciprocity game proposed for a directed cycle by Boyd and Richerson to the case of general networks. The model is not evolutionary and concerns the conditions under which the unanimity of cooperative players is a Nash equilibrium. I show that an abundance of triangles or other short cycles in a network does little to promote upstream reciprocity. Cooperation is less likely for a larger population size even if triangles are abundant in the network. In addition, in contrast to the results for evolutionary social dilemma games on networks, scale-free networks lead to less cooperation than networks with a homogeneous degree distribution.