Comparison of Oracles

Referee: [§3.2, Theorem 1] §3.2, Theorem 1 (deterministic case): the necessity direction claims that if dominance holds then the signaling functions admit simultaneous posterior matching. The argument appears to rely on constructing a game in which any mismatch in posteriors produces a strict equilibrium-outcome difference; however, the construction uses a specific payoff structure. It is unclear whether the same mismatch can be exploited in every game (including zero-sum or coordination games) without additional assumptions on equilibrium selection. A concrete counter-example or a fully general reduction would strengthen the claim.

Authors: The necessity direction of Theorem 1 proceeds by contrapositive. If the signaling functions fail to admit simultaneous posterior matching, we exhibit one specific game whose payoffs are constructed so that a posterior mismatch allows the second oracle to support a strict equilibrium outcome (e.g., a player’s preferred action) that the first oracle cannot replicate. Because dominance requires outcome-set inclusion in every game, failure in this single constructed game is sufficient to refute dominance; the argument does not claim or require that the mismatch affects outcomes in every conceivable game class such as zero-sum or coordination games. The payoffs are chosen to render the relevant equilibrium unique or strictly dominant, thereby sidestepping selection issues. We will add a short clarifying paragraph emphasizing the contrapositive structure and its independence from other game families. revision: partial

Referee: [§4.1, Definition 3 and Theorem 2] §4.1, Definition 3 and Theorem 2 (stochastic case): the common-knowledge component of the partition refinement is asserted to be sufficient for dominance. Because higher-order beliefs can affect equilibrium selection in incomplete-information games, it is necessary to verify that the common-knowledge refinement preserves the entire equilibrium correspondence, not merely the first-order posteriors. The current proof sketch does not explicitly address games in which the common-knowledge event itself is payoff-relevant.

Authors: We agree that an explicit verification that the common-knowledge refinement preserves the full equilibrium correspondence—including when the common-knowledge event is itself payoff-relevant—would strengthen the argument. The refinement operates at the level of the common-knowledge partition, which governs all higher-order beliefs and ensures that any equilibrium must be measurable with respect to that partition. In the revision we will expand the proof of Theorem 2 to include a direct argument (or a short example) showing that, when the common-knowledge event carries payoff consequences, the sets of inducible Nash equilibrium outcomes coincide under the two refined information structures. This extension will confirm that the characterization accounts for equilibrium selection effects arising from higher-order beliefs. revision: yes