The Lever Game is primarily played by flipping the lever against your opponent. It was invented by Teep (a.k.a Jon Evans). He has won the game the most, mainly due to the fact he invented it.
In order to win the lever game, you either:
(a) Flip the lever more than your opponent, without delaying, as in the case of Ravs when Rythian was speaking with Teep and Ravs beat Teep
(b) Spontaneously attack your opponent.
This is done mainly by Teep, either raging or simply doing so due to the rules.
It was created by Teep in Rythian's Minecraft series Road Trip, Part II. At the beginning of the episode Teep breaks down a lever from an unknown location somewhere in The Captive Creeper and starts flipping it on a table. Rythian exclaims over how random Teep is and how he's amazed Teep understands human speech. Later in the episode Teep starts his first opponent match with Ravs right outside the Crooked Caber, and proceeds to play with Ravs until he does his signature spontaneous attack after finishing the game. This was the first precedence and an example of how the game was to be played.
The link to the episode can be found here.