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A basic lesson from game theory is that strategic behavior often renders the equilibrium outcome inefficient. The recent literature of information design -- a.k.a. signaling or persuasion -- looks to improve equilibria by providing carefully-tuned information to players in order to influence their actions. Most previous studies have focused on the question of designing optimal signaling schemes. This work departs from previous research by considering a descriptive question and looks to quantitatively characterize the power of signaling (PoS), i.e., how much a signaling designer can improve her objective of the equilibrium outcome. We consider four signaling schemes with increasing power: full information, optimal public signaling, optimal private signaling and optimal ex-ante private signaling. Our main result is a clean and tight characterization of the additional power each signaling scheme has over its immediate predecessor above in the broad classes of cost-minimization and payoff-maximization games where: (1) all players minimize non-negative cost functions or maximize non-negative payoff functions; (2) the signaling designer (naturally) optimizes the sum of players' utilities. We prove that the additional power of signaling -- defined as the worst-case ratio between the equilibrium objectives of two consecutive signaling schemes in the above list -- is bounded precisely by the well-studied notion of the price of anarchy of the corresponding games. Moreover, we show that all these bounds are tight.