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Most algorithmic studies on multi-agent information design so far havefocused on the restricted situation with no inter-agent externalities; a fewexceptions investigated truly strategic games such as zero-sum games andsecond-price auctions but have all focused only on optimal public signaling.This paper initiates the algorithmic information design of both \emph{public}and \emph{private} signaling in a fundamental class of games with negativeexternalities, i.e., singleton congestion games, with wide application intoday's digital economy, machine scheduling, routing, etc.  For both public and private signaling, we show that the optimal informationdesign can be efficiently computed when the number of resources is a constant.To our knowledge, this is the first set of efficient \emph{exact} algorithmsfor information design in succinctly representable many-player games. Ourresults hinge on novel techniques such as developing certain "reduced forms" tocompactly characterize equilibria in public signaling or to represent players'marginal beliefs in private signaling. When there are many resources, we showcomputational intractability results. To overcome the issue of multipleequilibria, here we introduce a new notion of equilibrium-\emph{oblivious}hardness, which rules out any possibility of computing a good signaling scheme,irrespective of the equilibrium selection rule.

Algorithmic Information Design in Multi-Player Games: Possibility and Limits in Singleton Congestion