Tyrion visits the pyromancers/alchemists deep under the Red Keep. He manages not to get blown up. The pyromancers want to show off to King Joff but the last thing anyone wants is Joffrey playing with explosives.
Cleos Frey turns up with his Robb's peace terms. Tyrion thinks a bit of back and forth with messages relaying unacceptable terms will be an excellent time-waster. He tells Jacelyn to make sure Robb's men don't get to see what a hellhole the city is becoming and what the prophets are ranting about.
Cersei has found out Tyrion want Myrcella to marry the Dornishman Prince Doran, which means Maester Pycelle has been leaking secrets. Tyrion gets slapped about for his trouble, but at least he makes Cersei cry.
Strategy-wise, Tyrion thinks Robb can't march on King's Landing without taking Harrenhal first, because Tywin would come to the rescue otherwise. And Robb isn't strong enough to take Harrenhal.
