  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.
