To save people clicking (cause this is Reddit and even with sources, people be mad you don't just tell them):

The origins of The Game are uncertain. The most common hypothesis is that The Game derives from another mental game, Finchley Central. While the original version of Finchley Central involves taking turns to name stations, in 1976, members of the Cambridge University Science Fiction Society (CUSFS) developed a variant wherein the first person to think of the titular station loses. The game in this form demonstrates ironic processing, in which attempts to suppress or avoid certain thoughts make those thoughts more common or persistent than they would be at random.

How this became simplified into The Game is unknown; one hypothesis is that once it spread outside the Greater London area, among people who are less familiar with London stations, it morphed into its self-referential form. The creators of "LoseTheGame.net", a website which aims to catalogue information relating to the phenomenon, have received messages from multiple former members of the CUSFS commenting on the similarity between the Finchley Central variant and the modern Game. The first known reference to The Game is a blog post from 2002 – the author states that they "found out about it online about 6 months ago".

The Game is most commonly spread through the internet, such as via Facebook or Twitter, or by word of mouth.