Most academic papers are rejected by several journals before some journal finally accepts them. But a paper's rejection history is usually private; all readers know is where each paper was accepted.
Imagine a journal that published all its rejections, listing rejected authors, titles, and relevant dates. The possibility of embarrassment via appearing on such a list would "raise the bar" for authors, especially discouraging those who thought rejection more likely. It would also raise the bar for editors; readers could see how often rejected papers were accepted at equal or better journals, and potential authors could better evaluate their chances.
Since this signal of author and editor confidence would speak well of a journal, journals that did not publish rejections should look worse by comparison. Why then do no journals publish their rejections? Sporting contests publicly display losers; why not academic contests as well?
By the way, a new math journal publishes rejects only.
Added: most comments focus on the overall social effects; my puzzle is regarding individual incentives. People are usually eager to signal confidence in their abilities; why in this context do people avoid such signals?