Learning in Policy Transparency Games: Wedge Structure and Adaptive Certification

As learning agents are increasingly deployed in strategic environments, designers must choose whether to deploy transparent (auditable, committing) or opaque (flexible, reactive) policies. We formalize this through \emph{Policy Transparency Games} (PTGs): a two-stage model in which each agent first chooses transparency, then plays an underlying normal-form game. Our central theoretical contribution is a structural reduction: for two-player PTGs, all transparency incentives decompose into two per-player quantities, a leader wedge $L_i-N_i$ and a follower wedge $F_i-N_i$, whose signs characterize every equilibrium and cleanly separate selection-robust from selection-dependent conclusions. We then ask how a learning agent can discover this structure from interaction data. External-regret learners over the binary transparency action reach coarse correlated equilibrium, and our adaptive WEDGE-CERTIFY algorithm attains instance-dependent certificate complexity $\tilde O(\sigma^2(d_0^{-2}+d_{10}^{-2}+d_{01}^{-2}))$ with matching information-theoretic lower bounds. The wedge representation thus determines the learnability of endogenous algorithmic transparency.