 Solution

With equally likely codewords, the prior factor in the ideal-observer criterion is constant, so it agrees with maximum likelihood. On the binary symmetric channel,For $0<p<1/2$ this is strictly decreasing in Hamming distance, so the likelihood maximizers are precisely the nearest codewords. With the same tie convention all three decoders therefore agree. At $p=0$ every possible received word equals its transmitted codeword, and the rules agree on all positive-probability outputs; outputs having zero probability need no posterior assignment.