This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.
John von Neumann and John Nash built the branch of game theory most people mean by the term: players with defined payoffs, choosing strategies, converging or failing to converge on an equilibrium. John Conway built a different one almost by accident, while trying to understand the endgame of Go. Late-stage Go positions often break into several independent regions of the board where play barely interacts, and Conway, analyzing how the value of each region combines with the others, found that the combining itself obeyed rules that looked exactly like arithmetic on a new and much larger kind of number. He worked out the theory properly and published it in 1976 as On Numbers and Games, and the numbers themselves came to be called surreal numbers: a system vast enough to contain every real number, every ordinal number, and infinitely many infinitesimal and infinite quantities besides, all constructed from nothing but the recursive idea of a game position and its possible moves. This is combinatorial game theory, and it is a genuinely different animal from the von Neumann and Nash tradition. Von Neumann and Nash games typically involve simultaneous choices, incomplete information about the other player's intentions, and payoffs that can require compromise or bluffing to navigate. Conway games are sequential, perfect-information, no-chance contests like Go, chess endings, or Nim, where the entire question is which player, moving optimally, forces a win. The two traditions share a name and a founding motivation, understanding strategic play mathematically, and very little else about their machinery. That a Cambridge mathematician found a new kind of number by staring hard enough at a Go endgame is, on its own terms, one of the stranger origin stories in modern mathematics.