Clustering in large networks does not promote upstream reciprocity

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.

💡 Research Summary

In this paper the author investigates whether the structural property of clustering—i.e., the abundance of short cycles such as triangles—can promote cooperation based on upstream reciprocity (also called generalized reciprocity) in large social networks. Upstream reciprocity is an indirect‑reciprocity mechanism in which a player who has been helped by someone else helps a third party. Although laboratory experiments have repeatedly observed this behavior in humans and even in rats, most theoretical work suggests that upstream reciprocity can sustain cooperation only in small groups, when combined with other mechanisms, or when the underlying interaction graph contains very short cycles. Because empirical social networks are typically rich in triangles, it has been conjectured that real‑world networks might be conducive to upstream reciprocity.

To test this conjecture the author extends the classic Boyd‑Richerson model, originally defined on a directed cycle, to arbitrary directed (or undirected) weighted graphs. Each node can adopt one of two pure strategies: (i) a “generous cooperator” (GC) who donates a unit cost c to a downstream neighbor whenever it has received a donation in the previous time step, and (ii) a “classical defector” (CD) who never donates. In the GC state the flow of donations in the network reaches a steady state that coincides with the stationary distribution v of a simple random walk on the graph, satisfying v = v D⁻¹A where A is the adjacency matrix and D is the diagonal matrix of out‑degrees. The total amount a GC i gives to each downstream neighbor j per time step is N v_i A_{ij}/k^{out}_i, where N is the number of players.

The core analytical contribution is a derivation of the condition under which the unanimous GC profile is a Nash equilibrium, i.e., no single player can profit by deviating to CD. By computing the discounted sum of payoffs for a player i who switches to CD while all others remain GC, the author obtains a matrix‑geometric series that can be summed in closed form because the spectral radius of w (I − E_i) D⁻¹A is less than one (w∈