  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.
