How useful is adaptive action?

Channels with action-dependent states, as defined in [1], are considered. While [1] investigated the scenario of message dependent non-adaptive action sequences, this work focuses on adaptive action sequences, where the action is a strictly causal function of the message and the state sequences. The capacity of such a channel is characterized for the case when the state information is available non-causally at the channel encoder and it is shown that the adaptive action is not useful in increasing the capacity. The capacity of the action-dependent additive Gaussian channel, which was left open in [1], is then characterized by showing the equivalence of the current setting to the problem of the cooperative multiple access channel (MAC) with asymmetric state information at the encoders [2]. The problem setting is then extended to characterize the rate-distortion region of the source coding dual of the adaptive action-dependent channel coding setup, where a source is to be communicated with some fidelity to a decoder with adaptive action-dependent side information. In this setting adaptivity offers no help in enlarging the rate-distortion region.