We investigate the effects of update rules on the dynamics of an evolutionary game-theoretic model – the N-player evolutionary trust game – consisting of three types of players: investors (trusters), trustworthy trustees, and untrustworthy trustees. Interactions among players are constrained by local neighborhoods predefined by spatial or social network topologies. We compare different evolutionary update rules with behaviors that rely on the level of payoffs obtained by neighbors. In particular, we study the dynamics resulting from players using a deterministic rule (i.e., unconditional imitation with and without using a noise process induced by a voter model), a stochastic pairwise payoff-based strategy (i.e., proportional imitation), and stochastic local Wright-Fisher processes. We study these dynamics with different social network structures and varying levels of game difficulty. We see that there are significant differences on the level of promoted trust and global net wealth depending on the update rule. Under ‘harder’ game settings, rules based on unconditional imitation achieve the highest global net wealth in the population. We observe that there are key spatio-temporal correlations in the system for all rules. The update rules lead to the formation of fractal structures on a lattice, and low frequencies in the output signal of the system (i.e., long-term memory) when the rules are stochastic.