In Playing at the World, Jon Peterson provides a meticulously researched history of simulating wars, people, and fantastic adventures:
The key thread appears in the work of the younger Reiswitz, in the 1820s. He first introduced to wargaming two separate but intertwined components: referees and dice.
Reiswitz developed a game where verbal orders took the place of movements on a board, where a referee interpreted statements of intention from the players and converted them into results in the game world. This feedback loop, where the referee explains the state of the world and players then describe the actions they would like to attempt, is the fundamental innovation that underlies role-playing games. It bounced across languages and continents until it resurfaced in America in the work of Totten in the 1880s, which Twin Cities gamers later rediscovered and made part of their games in the late 1960s.
This achievement alone would be sufficient to earn a place in the pantheon of gaming gods, but the younger Reiswitz was also the first to grasp how statistics and probability could be combined to let dice resolve fictional events in a game. At his day job at the artillery ranges, he learned the differences in likelihood of striking targets with firearms at different ranges, and from those statistical models, he was able to assign a probability that die throws could resolve as game events. I believe this is where the fundamental principle of simulation was invented, and it was something then unprecedented in intellectual history. He also grasped that dice were a critical enabler for the referee as well, because dice are impartial: an omnipotent referee could always show an unconscious bias towards participants in the game, but dice kept the referee honest.
These two innovations walked hand in hand through the centuries right up to your table top.