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The actions of intelligent agents, such as chatbots, recommender systems, and virtual assistants are typically not fully transparent to the user. Consequently, using such an agent involves the user exposing themselves to the risk that the agent may act in a way opposed to the user's goals. It is often argued that people use trust as a cognitive shortcut to reduce the complexity of such interactions. Here we formalise this by using the methods of evolutionary game theory to study the viability of trust-based strategies in repeated games. These are reciprocal strategies that cooperate as long as the other player is observed to be cooperating. Unlike classic reciprocal strategies, once mutual cooperation has been observed for a threshold number of rounds they stop checking their co-player's behaviour every round, and instead only check with some probability. By doing so, they reduce the opportunity cost of verifying whether the action of their co-player was actually cooperative. We demonstrate that these trust-based strategies can outcompete strategies that are always conditional, such as Tit-for-Tat, when the opportunity cost is non-negligible. We argue that this cost is likely to be greater when the interaction is between people and intelligent agents, because of the reduced transparency of the agent. Consequently, we expect people to use trust-based strategies more frequently in interactions with intelligent agents. Our results provide new, important insights into the design of mechanisms for facilitating interactions between humans and intelligent agents, where trust is an essential factor.