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?
But in a sports contest, especially ones involving teams, many of those doing the affiliating are affiliating with the losing team, and continue to do so, usually, even if the team loses. They are paying to attend to see their team compete with the hope that they are going to win, or at least play well.
Those reading journal articles, especially those purchasing subscriptions to the journal, are doing so to read the articles published in the journal, or perhaps to affiliate with the journal. But do these people think about "affiliating" with the authors or titles of papers rejected for publication? Certainly not in the same way that fans affiliate with a losing team, or in the case of more individualized sports such as tennis or golf, with a losing player.
Barkley, I only suggested publishing titles/authors/dates. You say for sports "the contest ... is the activity" but "for academic .. the bottom line is ... the contents of the ... paper" but to me that just seems another way to rephrase the question. I see both industries as ultimately allowing customers to affiliate with certified impressive folks, so the issue is about different ways to sort/certify impressiveness.