Johnson and his Oxford colleagues David Lamper, Paul Jefferies, Michael Hart and Sam Howison have further studied a binary aspect of the game: the up-and-down movements in a market irrespective of magnitude, which is simpler and computationally practical.

• 强生和他的牛津同事兰珀、杰夫里兹、哈特及豪伊森进一步研究这个游戏的二元观点，也就是只管市场里的上下移动，而不管移动的大小，那是比较简单并且可以计算的部分。