For independent Gaussian signals, a stronger privacy requirement pushes the encoder toward sending X directly, which improves the decoder's accuracy even though the encoder's own objective worsens.
Strategic information transmission against malicious eavesdropper and semi-honest estimator: A tripartite game analysis,
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On the Interplay of Privacy, Persuasion and Quantization
For independent Gaussian signals, a stronger privacy requirement pushes the encoder toward sending X directly, which improves the decoder's accuracy even though the encoder's own objective worsens.