Joffrey doesn't need convincing. If you remember Littlefinger conversation with Margaery Tyrell in a tent after Renly's murder she told him that she 'wants to be The Queen'. And Littlefinger also met Tywin in Harrenhall where Arya was serving wine for them. We saw a start of that conversation where Tywin asked Littlefinger if Margaery still wants to be queen... So all in all this scene is just an act in order to put Sansa aside with some dignity to the crown as all the things have already been settled secretly by houses Lannister and Tyrell. Tywin organised this marriage in order to make alliance with them and get their help against at least Stannis forces.
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There may be a 2nd meaning to song of ice and fire.
Joffrey doesn't need convincing. If you remember Littlefinger conversation with Margaery Tyrell in a tent after Renly's murder she told him that she 'wants to be The Queen'. And Littlefinger also met Tywin in Harrenhall where Arya was serving wine for them. We saw a start of that conversation where Tywin asked Littlefinger if Margaery still wants to be queen... So all in all this scene is just an act in order to put Sansa aside with some dignity to the crown as all the things have already been settled secretly by houses Lannister and Tyrell. Tywin organised this marriage in order to make alliance with them and get their help against at least Stannis forces.