  The trust game, derived from an economics experiment, has recently attracted
interest in the field of evolutionary dynamics. In a recent version of the
evolutionary trust game, players adopt one of three strategies: investor,
trustworthy trustee, or untrustworthy trustee. Trustworthy trustees enhance and
share the investment with the investor, whereas untrustworthy trustees retain
the full amount, betraying the investor. Following this setup, we investigate a
two-player trust game, which is analytically feasible under weak selection. We
explore the evolution of trust in structured populations, factoring in four
strategy updating rules: pairwise comparison (PC), birth-death (BD), imitation
(IM), and death-birth (DB). Comparing structured populations with well-mixed
populations, we arrive at two main conclusions. First, in the absence of
untrustworthy trustees, there is a saddle point between investors and
trustworthy trustees, with collaboration thriving best in well-mixed
populations. The collaboration diminishes sequentially from DB to IM to PC/BD
updating rules in structured populations. Second, an invasion of untrustworthy
trustees makes this saddle point unstable and leads to the extinction of
investors. The 3-strategy system stabilizes at an equilibrium line where the
trustworthy and untrustworthy trustees coexist. The stability span of
trustworthy trustees is maximally extended under the PC and BD updating rules
in structured populations, while it decreases in a sequence from IM to DB
updating rules, with the well-mixed population being the least favorable. This
research thus adds an analytical lens to the evolution of trust in structured
populations.
