The escalation die represents a bonus to attacks as the fight goes on.
At the start of the second round, the GM sets the escalation die at 1. Each PC gains a bonus to attack rolls equal to the current value on the escalation die. Each round, the escalation die advances by +1, to a maximum of +6.
Monsters and NPCs do not add the escalation die bonus to their attacks (there are exception, dragons get the bonus for example).
The escalation die also can be used to determine when certain things trigger during a round. For example, lava vents erupt when the escalation die is even. Some monsters might have abilities that vary based on the escalation die such as the following: Marillith demons crit range increases by the escalation die value, frost giants have a chance of preventing the escalation die from increasing, dragons not only get to add the escalation die to their attacks like PCs but roll a d6 each turn and if they roll less than the current escalation die value get an extra action that turn.
If the GM judges that the characters are avoiding conflict rather than bringing the fight to the bad guys, the escalation die doesn’t advance. If combat virtually ceases, the escalation die resets to 0.
Notes about why:
There are many items from 13th Age that we are incorporating, one of them is the escalation die. Though we will end up using the Escalation die mechanically, the important bit is the spirit of it. In 13th Age battles are front-loaded. If you’re going to lose, you’re probably going to lose in the first few rounds and then you move on. If you survive those first few rounds, the escalation die tips the fight heavily in your favor so that you don’t spend two hours mopping up mooks and lets you move on. 13th Age really wants you to keep going. It’s a change of expectation that means what we’re about to do is going to make far more sense than it would in 3.x/4E. This may not be an issue in 5e, but the spirit is a good one and the mechanical concept is a nice one for adding other effects to the game. We’ll try it out and see how it goes.