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：信息透明度历来被视为任何组织或系统中有效治理、问责与道德行为的关键要素。然而，一个自然的问题随之而来：*透明的代价或收益究竟为何*？因为透明度可能对信息结构施加额外约束，从而降低信息提供者的操作空间。本文通过比较两种代表性信息结构——公开劝说模型与隐秘信号传递模型——下的完美贝叶斯均衡收益，提出并定量研究了战略信息披露中的*透明代价*（Price of Transparency, PoT）。PoT定义为隐秘互动与公开互动中收益结果的比率。作为主要贡献，本文发展了一种双层双线性规划方法，称为$Z$-规划，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，我们证明选择透明信息结构始终符合信息提供者的利益，因为$0\leq \textrm{PoT}\leq 1$。上界在任何严格贝叶斯后验竞争博弈（零和博弈为其特例）中均可达到。对于连续博弈，PoT仍以$1$为上界，但可任意接近$0$，表明下界的紧致性。这一紧致下界说明，缺乏透明度可能导致信息提供者遭受重大损失。我们通过二次博弈及数值算例验证了上述发现。