The short story was printed in an anthology, possibly between the 60s and 70s, and by a mainstream author in a wacky mood.
The protagonist is constantly upstaged by his neighbor. A nuclear war occurs and humanity is decimated and devolves. The protagonist's descendant is jealous of the bigger, smoother sunning rock used by "other me".
The story is told in a series of narrations beginning in the past, told through the present and into the future described above.