I don't think it was explicitly a double game, more of a look the way other way with a few intelligence leaks here and there. The drone strikes happened in FATA because of PA inaction, troops should have been moved as soon as the invasion of Afg was announced.
Pakistan pussyed out the fight and icing on the cake was OBL in Pakistan with a big fat $100 billion loss. Could have all been avoided.
No doubt Pakistan miscalculated US intention in Afghanistan. It would have served our interest far more had we sent troops along with NATO paid for by the US. Would have opened the avanue for significant US-Pakistan cooperation in the fields of economy, Defence tech, foreign policy ,etc... Could have even been used as a leverage vis via Kashmir.
One thing most people forget is the US invasion of Iraq in 2003 and its effects on the Afghan war. This is really when the US lost its narrative. If I were a Pak General/ISI post 2003 invasion of Iraq, id be scared that Pakistan would be on the list. If you may recall also post 2003-2010 we saw a huge deal of misinformation come up in Pakistan, with claims that the US is sponsoring BLA and hopes to break Pakistan up into 4 pieces by 2015 but none of this ever occured. US generally was supportive of Pakistan, even when many of its troops were targetted by the same taliban. US personel were also targets throughout the 2000s in Pakistan with many US assets targetted.
This was a complete U-turn from the historic perception the pakistani public had of the US up until that point. But a lot of Pakistans undercover support for the Taliban could be pointed back to fears due to the US invasion of Iraq in 2003.