  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 $n\to\infty$.
