Out-earning or outnumbering: Two routes to cooperation in the fixed-role trust game on structured populations

Trust underlies cooperation, but sustaining it is individually costly and difficult under rational self-interest. In the trust game, a trustor invests part of an endowment and a trustee returns a share of the multiplied proceeds. Under fixed roles on a square lattice, agents play only with orthogonal neighbors of opposite-roles, but mimic diagonal neighbors of the same-role, decoupling interaction from imitation networks. Trust that collapses in well-mixed populations can instead co-evolve with trustworthiness on the lattice within a narrow range of the return parameter under strong selection. Comparing update rules uncovers two distinct routes to cooperation. First, pairwise imitation depends on reciprocators (that is, reciprocating trustees) outperforming betrayers through local assortment, forming clusters of trustworthy trustees around generous trustors. In contrast, a neighborhood rule allows the local majority of reciprocators convert an isolated betrayer regardless of the payoff, outnumbering betrayers rather than out-earning them. This collective-recruitment route sustains trust-reciprocate interactions over a wider range of the return parameter. The two routes differ, however, in what they require: pairwise spread needs only cross-role assortment and still produces cooperation on a geometry-free degree-regular network, whereas collective recruitment does not survive that change. Degree heterogeneity adds no route of its own, and whether it leaves cooperation standing depends on which role carries it and on how payoffs are aggregated. Of the structures examined here, only the lattice supports both routes.