Strategic Quantization

This paper is concerned with the quantization setting where the encoder and the decoder have misaligned objectives. While the unconstrained variation of this problem has been well-studied under the theme of information design problems in Economics, the problem becomes more appealing and relevant to engineering applications with a constraint on the cardinality of the message space. We first motivate the problem via a toy example demonstrating the intricacies of the strategic variation of the quantization problem, explicitly showing that the quantization resolution can change the nature of optimal encoding policy and the iterative optimization of the decoder and the encoder mappings may not converge to a local optimum solution, in sharp contrast with the classical quantization problem that admits the Lloyd-Max method as a local optimum solution. As a remedy, we develop a gradient-descent based solution. We analyze the poor local optimal optima issues associated with the optimization method and show that even for well-behaving sources, such as Gaussian, there are multiple local minima, depending on the distortion measures chosen, in sharp contrast with the classical quantization. We finally present numerical results obtained via the proposed algorithm that suggest their validity and demonstrate features of the strategic version of the quantization problem that differentiate it from its classical counterpart.