  We introduce the zero-sum game problem of soft watermarking: The hidden
information (watermark) comes from a continuum and has a perceptual value; the
receiver generates an estimate of the embedded watermark to minimize the
expected estimation error (unlike the conventional watermarking schemes where
both the hidden information and the receiver output are from a discrete finite
set). Applications include embedding a multimedia content into another. We
consider in this paper the scalar Gaussian case and use expected mean-squared
distortion. We formulate the resulting problem as a zero-sum game between the
encoder & receiver pair and the attacker. We show that for the lin- ear
encoder, the optimal attacker is Gaussian-affine, derive the optimal system
parameters in that case, and discuss the corresponding system behavior. We also
provide numerical results to gain further insight and understanding of the
system behavior at optimality.
