ATAMI, Japan—This resort town, once popular with honeymooners, is turning to a new breed of romance seekers—virtual sweethearts. Since the marriage rate among Japan's shrinking population is falling and with many of the country's remaining lovebirds heading for Hawaii or Australia's Gold Coast, Atami had to do something. It is trying to attract single men—and their handheld devices. In the first month of the city's promotional campaign launched July 10, more than 1,500 male fans of the Japanese dating-simulation game LovePlus+ have flocked to Atami for a romantic date with their videogame character girlfriends. The men are real. The girls...