A general formula for capacity of channels with action-dependent states

Weissman introduced a channel coding problem for channels with action-dependent states. In this coding problem, there are two encoders and a decoder. One encoder outputs an action that affects states of the channel. Then, the other encoder encodes a message by using the channel state, and its codeword is fed into the channel. The decoder receives a noisy observation of the codeword, and reconstructs the message. For this coding problem, Weissman showed the capacity when states and the channel are stationary memoryless. In this paper, we show a general formula of the capacity when states and the channel may not be stationary memoryless, which is expressed by mutual information spectrum-sup/inf proposed by Verdú and Han. Our general formula coincides with the capacity derived by Tan when actions cannot affect states of channels. We also show that the capacity for nonstationary memoryless channels can be expressed by using ordinary mutual information.