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 $n\to\infty$.

翻译：对于任意有限离散信源，前缀码 $C_1$ 相对于前缀码 $C_2$ 的竞争优势定义为 $C_1$ 生成比 $C_2$ 更短码字的概率减去 $C_2$ 生成比 $C_1$ 更短码字的概率。若一个前缀码对所有其他前缀码具有非负的竞争优势，则称该前缀码在竞争意义下最优。1991年，Cover证明霍夫曼编码对所有二元信源（即符号概率为2的负整数次幂的信源）均具有竞争最优性。本文证明如下渐近逆命题：随着信源规模增大，随机选择的非二元信源的霍夫曼编码具有竞争最优性的概率趋近于零。此外还证明：(i) 对任意非二元信源，霍夫曼编码相较于Shannon-Fano编码具有正的竞争优势；(ii) 对任意信源，任意前缀码相对于霍夫曼编码的竞争优势严格小于 $\frac{1}{3}$；(iii) 对每个整数 $n>3$，存在规模为 $n$ 的信源及某个前缀码，使其相对于霍夫曼编码的竞争优势可任意接近 $\frac{1}{3}$；以及 (iv) 对每个正整数 $n$，存在规模为 $n$ 的信源及某个前缀码，使其相对于Shannon-Fano编码的竞争优势随 $n\to\infty$ 可任意接近 $1$。