When Nature Whispers to the Opponent: Information Leakage in Robust Markov Games

Robust Markov games contain two adverse entities: a strategic opponent and nature, which governs uncertain transitions. Fusing them into a single adversary can change the information structure: if nature observes a hidden mode, the fused opponent may condition on information unavailable to the modeled opponent. We formalize the intended counterfactual by relaxing only the opponent's information while preserving the ambiguity family, transition dynamics, timing, and one complete nature plan. A four-value hierarchy separates this information effect from stronger relaxations that reoptimize continuation plans across public actions or private histories. For each fused profile, we construct a legal public-history shadow with the same nature plan; the KL divergence from the fused law to its shadow equals the cumulative conditional mutual information carried by opponent actions, yielding information-budget bounds and a zero-leakage exactness criterion. Finite-game, learning, pursuit-evasion, and cyber-defense experiments validate the distinction. In cyber defense, fusion raises a policy's worst-case safety cost from $1.244$ to $3.000$, reversing safe policy selection at threshold $1.4$. For learned attackers, rollout estimates of leakage and public-shadow values match enumeration, while leakage regularization yields a value-leakage tradeoff.