On Saturday evening, Dov Bergwerk arrived at the Avis branch on West 76th Street and Broadway. Accompanied by his wife Ruth, the Bergwerks were planning to join friends for dinner in Westchester. Mr. Bergwerk, a senior vice president and general corporate counsel at the Israeli pharmaceutical giant Teva, got out his driver’s license, reservation number and “Wizard†loyalty card – he’s rented from Avis dozens of times before – and anticipated the usual smooth transition into a nondescript mid-sized sedan. That’s when the trouble started. A reservation agent named Angelline declined to honor Mr. Bergwerk’s reservation, saying that it was...