In 2003,Eric Hautemont was prowling the corridors of the World Boardgaming Championships, looking for a game.

He and two partners had founded an old-fashioned cardboard, dice and wood board game company in Silicon Valley called Days of Wonder. Eventually, he and co-founder Mark Kaufmann sat down to play a prototype of a train-themed game—at the time, just a handmade piece of cardboard with line drawings. The game, made by a well-known designer named Alan R. Moon, had three basic rules and a simple goal: to collect the most...