  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.
