Question

why does the author brood over a lost cause

Answer 1

The author makes several attempts to remind Todd about the dollar he owed. He meets Todd almost every day in the Club but Todd never mentions about his debt and even says that he no longer remembers much about his Bermuda trip. The author feels exasperated. He feels that his dollar is lost forever.
In desperation he writes off the dollar. He writes Todd's name in the list of the people who have similarly forgotten to repay their one dollar loans.