In Bayesian online selection, a decision-maker observes a sequence of stochastic rewards and must immediately and irrevocably accept or reject each realized value. The rewards come from known distributions, referred to as their identities. Whereas classical prophet inequalities compare online algorithms to the offline optimum, recent work studies information gaps relative to the optimal online algorithm, which knows the identities' arrival order. We introduce a new restriction motivated by identity-independent decision-making: the decision-maker observes the current value but learns its identity only after deciding whether to accept it. We call such algorithms identity-truthful. We define the identity-truthfulness gap as the optimal worst-case approximation achievable by identity-truthful algorithms relative to the optimal online benchmark. This gap lies between the order-competitive ratio [Ezra et al. 2023] and the identity-blindness gap [Ezra et al. 2024]. We design an algorithm establishing that the identity-truthfulness gap is strictly greater than $0.5$, thereby separating it from the identity-blindness gap, which is exactly $0.5$. We complement this result with an upper bound of $0.81$ on the identity-truthfulness gap, which is strictly lower than the known upper bound of $0.829$ on the order-competitive ratio [Chen et al. 2024]. Finally, we study the relationship between identity-truthful algorithms and pricing mechanisms. Motivated by questions on the power of posted prices in prophet inequalities [Duetting et al. 2020], we show that identity-truthfulness breaks the usual connection between online algorithms and pricing: there exists an instance for which the optimal identity-truthful algorithm cannot be implemented by any pricing mechanism.

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