Causal feedback does not increase finite DMC capacity under average cost

For each time $t$ and each positive-probability history $h=y^{t-1}$, the causal encoder and memoryless channel give the finite Markov chain $U\to X_t\to Y_t$. By chain rule [F1] and data processing [F2], $I(U;Y^n)\le {\sum }_{t=1}^n{\mathbb{E}}_{Y^{t-1}}{C}_I(\mathbb{E}[c(X_t)\mid Y^{t-1}])$, applying the definition of $C_I$ to each conditional input law, whose cost is at least $c_{\min }$.

Give a uniform random index $T\in [n]$ independent of the code and set $V=\mathbb{E}[c(X_T)\mid T,Y^{T-1}]$. Then $V\ge c_{\min }$, $\mathbb{E}V=n^{-1}{\sum }_t\mathbb{E}c(X_t)\le \mathrm{\Gamma }$, and $C_I(V)$ is integrable by [F3]. Applying [F4] and monotonicity in [F3] to step 1.1 yields $I(U;Y^n)\le n\mathbb{E}{C}_I(V)\le n{C}_I(\mathbb{E}V)\le n{C}_I(\mathrm{\Gamma })$.

If $M=1$, the rate is zero. If $M\ge 2$, the relaxed Fano bound [F5] and the deterministic decoder imply $(1-e)\ln M\le I(U;Y^n)+\ln 2\le n{C}_I(\mathrm{\Gamma })+\ln 2$. Thus reliable feedback rates are at most $C_I(\mathrm{\Gamma })=C(\mathrm{\Gamma })$ by [F6]. Every ordinary feasible code is a feedback code whose encoders ignore past outputs, so $C_{\mathrm{f}\mathrm{b}}(\mathrm{\Gamma })\ge C(\mathrm{\Gamma })$; the reverse inequality follows from the converse.