In decentralized decision-making problems, communicating agents choose their actions based on locally available information and knowledge about decision rules or strategies of other agents. In this work, we consider a strategic communication game between an informed encoder and two decoders communicating via a Gray-Wyner network. All three agents are assumed to be rational and endowed with distinct objectives captured by non-aligned cost functions. The encoder selects and announces beforehand the compression scheme to be implemented. Then, it transmits three signals: a public signal, and a private signal to each decoder inducing a Bayesian game among the decoders. We are interested in designing an achievable coding scheme that minimizes the encoder’s long run cost function subject to the challenges imposed by the Gray-Wyner network.

We study the multi-user Bayesian persuasion game between one encoder and two decoders, where the first decoder is better informed than the second decoder. We consider two perfect links, one to the first decoder only, and the other to both decoders. We consider that the encoder and both decoders are endowed with distinct and arbitrary distortion functions. We investigate the strategic source coding problem in which the encoder commits to an encoding while the decoders select the sequences of symbols that minimize their long-run respective distortion functions. We characterize the optimal encoder distortion value by considering successive refinement coding with respect to a specific probability distribution which involves two auxiliary random variables, and captures the incentive constraints of both decoders.