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.

翻译：信息透明度一直被视为任何组织或系统中有效治理、问责和道德行为的关键要素。然而，一个自然的问题随之而来：*透明化的成本或收益是什么*？因为人们可能怀疑透明度会对信息结构施加额外约束，从而降低信息提供者的可操作性。本研究通过比较两种代表性信息结构——公开说服模型与隐蔽信号模型——下的完美贝叶斯均衡收益，提出并定量探讨了战略信息披露中的*透明代价*（Price of Transparency, PoT）。PoT被定义为隐蔽互动与公开互动中收益结果的比率。作为主要贡献，本文开发了一种名为$Z$-规划的双层-双线性规划方法，用于求解具有有限状态和行动的动态不完全信息博弈的非退化完美贝叶斯均衡。利用$Z$-规划，我们证明信息提供者始终有动机选择透明信息结构，因为$0\leq \textrm{PoT}\leq 1$。该上界在任何严格贝叶斯后验竞争博弈中均可达到，零和博弈为其特例。对于连续博弈，PoT仍以$1$为上界，但可任意接近$0$，表明下界的紧性。这一紧下界表明，缺乏透明度可能导致提供者的显著损失。我们通过二次博弈和数值算例验证了上述发现。