Department of Mathematical Informatics, The University of Tokyo, Bunkyo, Tokyo, Japan.
PLoS One. 2011;6(10):e25190. doi: 10.1371/journal.pone.0025190. Epub 2011 Oct 5.
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.
上行互惠（也称为广义互惠）是一种假设的合作机制，用于解决社会困境问题，即在他人帮助自己的情况下，玩家会帮助他人。它是一种间接互惠。尽管上行互惠在实验中经常被观察到，但大多数理论表明，只有当玩家形成短循环（如三角形）时，它才起作用，这意味着人口规模较小，或者当它与其他单独促进合作的机制结合使用时。人们期望已知充满三角形和其他短循环的真实社交网络能够适应上行互惠。在这项研究中，我将 Boyd 和 Richerson 提出的针对有向循环的上行互惠博弈扩展到一般网络的情况。该模型不是进化模型，而是涉及到在何种条件下合作玩家的一致性是纳什均衡。我表明，网络中三角形或其他短循环的大量存在对促进上行互惠作用不大。即使网络中存在大量三角形，人口规模较大也会降低上行互惠的可能性。此外，与网络上进化的社会困境博弈的结果相反，无标度网络导致的合作比具有均匀度分布的网络要少。
