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How does the brain make decisions in sequential, multi-step tasks? In principle, evaluating choice options in such tasks requires simulating each sequence of states that might follow a choice and averaging the summed rewards along each sequence. In practice, such “model-based” approaches are prohibitively expensive and it is widely thought that the brain often relies on “model-free” approaches, which store pre-computed estimates of the cumulative rewards that will follow each choice. Although a prominent theory links the learning of these estimates to the action of mid-brain dopamine neurons on striatum, it is known that model-free approaches are too inflexible to explain the behavioral flexibility of humans and animals in tasks that require them to compute new reward predictions after learning that the task’s rewards have changed.