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Speakers communicate to influence their partner's beliefs and shape their actions. Belief- and action-based objectives have been explored independently in recent computational models, but it has been challenging to explicitly compare or integrate them. Indeed, we find that they are conflated in standard referential communication tasks. To distinguish these accounts, we introduce a new paradigm called signaling bandits, generalizing classic Lewis signaling games to a multi-armed bandit setting where all targets in the context have some relative value. We develop three speaker models: a belief-oriented speaker with a purely informative objective; an action-oriented speaker with an instrumental objective; and a combined speaker which integrates the two by inducing listener beliefs that generally lead to desirable actions. We then present a series of simulations demonstrating that grounding production choices in future listener actions results in relevance effects and flexible uses of nonliteral language. More broadly, our findings suggest that language games based on richer decision problems are a promising avenue for insight into rational communication.