jimmuller 

Interesting. Ask yourself what would be different if the middle pulley wasn't there. The chain would run directly from the idler pulley to the tension pulley and would thus be a bit shorter. The wrap length around the idler would be less, but that wouldn't normally be a critical parameter. (More on that in a moment.) Note that the relative positions of the three pulleys are fixed. So as the cage pivot moves, the total chain take-up is determined strictly by the position of the tension pulley. The middle pulley contributes nothing to the difference between maximum and minimum chain take-up.

From this vantage point it looks like the middle pulley does nothing except reduce the possibility of the chain jumping off the idler pulley, not terribly likely, but maybe this increases its ability to force the chain sideways against a recalcitrant cog.