We study the problem of forecasting for an arbitrary number of downstream agents with unknown objectives, each of whom best responds to the forecaster's predictions. We seek a single forecaster that guarantees sublinear swap regret for all downstream agents simultaneously. For two-dimensional outcome spaces, we give a polynomial time algorithm that guarantees $\tilde{O}(\sqrt{kT})$ swap regret for any downstream agent with $k$ actions. This improves over the previously known bound of $\tilde{O}(kT^{5/8})$ and avoids the exponential in $T$ runtime of prior algorithms in this setting. Our algorithm extends nicely to other low dimensional environments, retaining $\tilde{O}(\sqrt{T})$ downstream swap regret while the exponent of $k$ in the regret bound and the exponent of $T$ in the running time both grow with dimension. For arbitrary dimension $d$, we give a forecasting algorithm that guarantees $\tilde{O}(d\sqrt{kT})$ swap regret, assuming the forecaster knows an upper bound $k$ on the number of actions available to any downstream agent, albeit with a much longer runtime. This improves upon previous high dimensional guarantees that had $\tilde{O}(T^{2/3})$ dependence and required additional behavioral assumptions.

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