Competitive Advantage of Huffman and Shannon-Fano Codes

For any finite discrete source, the competitive advantage of prefix code $C_1$ over prefix code $C_2$ is the probability $C_1$ produces a shorter codeword than $C_2$, minus the probability $C_2$ produces a shorter codeword than $C_1$. For any source, a prefix code is competitively optimal if it has a nonnegative competitive advantage over all other prefix codes. In 1991, Cover proved that Huffman codes are competitively optimal for all dyadic sources, namely sources whose symbol probabilities are negative integer powers of $2$. We prove the following asymptotic converse: As the source size grows, the probability a Huffman code for a randomly chosen non-dyadic source is competitively optimal converges to zero. We also prove: (i) For any non-dyadic source, a Huffman code has a positive competitive advantage over a Shannon-Fano code; (ii) For any source, the competitive advantage of any prefix code over a Huffman code is strictly less than $\frac{1}{3}$; (iii) For each integer $n>3$, there exists a source of size $n$ and some prefix code whose competitive advantage over a Huffman code is arbitrarily close to $\frac{1}{3}$; and (iv) For each positive integer $n$, there exists a source of size $n$ and some prefix code whose competitive advantage over a Shannon-Fano code becomes arbitrarily close to $1$ as $n\to\infty$.