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Title: Learning, Fast and Slow

Abstract: As learning agents are increasingly deployed in strategic environments, the question of how to strategically choose a learning method becomes economically relevant. We introduce a meta-game in which decision makers select learning rates in continuous-time projected gradient dynamics and evaluate payoffs along the entire trajectory of the resulting inner game, not merely at its limit.

We use $2 \times 2$ games to map the strategic considerations that arise in choosing the speed of learning. Three game classes produce three qualitatively distinct phenomena. In dominance-solvable games, faster learning is unambiguously beneficial under strategic complementarity, but best responses become non-monotonic under strategic substitutability, so that both players may optimally moderate their speeds. In coordination games, the ratio of learning rates governs basins of attraction, enabling equilibrium selection but introducing payoff discontinuities that cause standard existence results to fail. In zero-sum games, each player has a strictly monotonic incentive to increase their learning rate, generating an arms race with no equilibrium when rates are unbounded. Finally, we show that near any convergent Nash equilibrium every player strictly prefers to learn faster, implying that the richer phenomena above are driven by behaviour away from equilibria.