Information embedding on actions

The problem of optimal actuation for channel and source coding was recently formulated and solved in a number of relevant scenarios. In this class of models, actions are taken either to acquire side information in an efficient way for source coding, or to control or probe effectively the channel state for channel coding. In this paper, the problem of embedding information on the actions is introduced by considering the presence of an additional decoder that observes only a function of the actions. For the source coding model, this decoder wishes to reconstruct a lossy version of the source being transmitted over the point-to-point link, while, for the channel coding problem, the decoder wishes to retrieve a portion of the message conveyed over the link. In both cases, single-letter performance characterizations are provided for various special cases, along with specific examples.