On the Price of Transparency: A Comparison between Overt Persuasion and Covert Signaling

Transparency of information disclosure has always been considered an instrumental component of effective governance, accountability, and ethical behavior in any organization or system. However, a natural question follows: \emph{what is the cost or benefit of being transparent}, as one may suspect that transparency imposes additional constraints on the information structure, decreasing the maneuverability of the information provider. This work proposes and quantitatively investigates the \emph{price of transparency} (PoT) in strategic information disclosure by comparing the perfect Bayesian equilibrium payoffs under two representative information structures: overt persuasion and covert signaling models. PoT is defined as the ratio between the payoff outcomes in covert and overt interactions. As the main contribution, this work develops a bilevel-bilinear programming approach, called $Z$-programming, to solve for non-degenerate perfect Bayesian equilibria of dynamic incomplete information games with finite states and actions. Using $Z$-programming, we show that it is always in the information provider's interest to choose the transparent information structure, as $0\leq \textrm{PoT}\leq 1$. The upper bound is attainable for any strictly Bayesian-posterior competitive games, of which zero-sum games are a particular case. For continuous games, the PoT, still upper-bounded by $1$, can be arbitrarily close to $0$, indicating the tightness of the lower bound. This tight lower bound suggests that the lack of transparency can result in significant loss for the provider. We corroborate our findings using quadratic games and numerical examples.